Creative Teaching: Heritage of R. L. Moore

by D. Reginald Traylor

With: William Bane Madieline Jones c D. Reginald Traylor, 1993

Contents

I 5

1 The Early Years 7

2 The Graduate and Post Graduate Years 45

3 The Formative Years 63

4 Return to Texas 85

5 The Mature Years: 1930-1953 103

6 The Years of “Modiﬁed Service” 125

II 171

7 The Doctoral Students of Robert Lee Moore. 173

8 Publications of Robert Lee Moore 175

9 Academic Descendants of R. L. Moore and Individual Pub- lications 181

i ii PREFACE

The motives underlying an undertaking resulting in such a compilation as included herein were of a mixed variety. There definitely is represented a deep respect for the genius of Robert Lee Moore and information, as offered, testifies in part to that genius. Beyond that, some distinct feeling has been developing that far too many teaching techniques have been assigned the label “The Moore Method.” Thus, some effort is made to describe that method, as Moore employed it. Of fundamental concern has been those circumstances which led to Moore’s retirement at the University of Texas. Is it true that certain methods of teaching, no matter how effective, simply can not exist in today’s major universities? Is it true that university enrollments can climb to such a number that certain teaching practices simply must be discarded, no matter how effective they may be? Is it rue that friction between faculty members is assured once one of them begins teaching effectively by the Moore method and the other does not teach after that fashion? Is there any realistic way to expose the Moore method to a non-mathematical administrator so that he can sense its worth without having actually experienced it himself? For these reasons, and others, an attempt is made to present some display of Moore’s teaching, along with a description of the settings and times of his efforts. Hopefully, those, who never experienced such teaching, might gain some appreciation for its worth. This treatise is in two parts. The first deals with a chronological treatment of Moore’s academic activities. Robert Lee Moore is thought by some to be a teacher whose effectiveness has been exceeded by no other. He taught the “doing” of mathematics; he caused students to develop their powers of rational thought. The method of teaching which he developed has come to be known as “The Moore Method.” A description of that method is offered herein by way of comments of his students, comments of others who knew of that method, and by a rather complete listing of those people who earned doctorates in mathematics and who are among his academic descendants, along with a bibliography for each. It is hoped

1 2 that such evidence, as provided in this treatise will offer some support to those who can teach effectively by the Moore method. Simultaneously, those students who disparage that approach should have some opportunity to grasp the effectiveness offered by that method. To those who wish to teach after that fashion, perhaps it will be suggested herein that this is a very difficult manner with which to teach. A person employing the Moore method must spend extreme amounts of time pondering his class, fitting his subject matter to his students and them to it, so that their growth in rational power will be successful. Particular attention is given to Moore’s teaching and academic philosophies in the last two chapters of the first section. Any university teacher is herewith invited to compare his own worth and academic values with those of Professor Moore. It should be rather clear that those, who argue that a researcher can not be a good teacher, must contemplate the example of Robert Lee Moore. Moreover, those who argue that they are good teachers, though they have no research interests or accomplishments, should feel honor bound to offer, as an example, at least one of their past students about whom, it very probably is true, did realize success in some substantial endeavor but very probably would not have if deprived of that particular teacher. The second part of the treatise resulted from a Master’s thesis effort of W. Bane. Just as one might argue that Robert Lee Moore may have made substantial contribution, though perhaps not as he did, had he not been influenced by E. H. Moore or G. B. Halsted, so might one say that many mathematicians and their mathematics would not have occurred had Robert Lee Moore not functioned as he did. W. Bane began an undertaking of identifying academic descendants of Robert Lee Moore, along with their publications. This effort was continued by M. Jones, resulting in an extensive list of mathematicians with individual bibliographies. Over five hundred academic descendants were identified. Extreme care was taken to assure accuracy. Much information was gained by conversation and by correspondence. In some instances there was real question as to who actually performed the direction of a student. Infrequently there were those who perhaps directed doctoral students, and were themselves “descendants” of Moore, who failed to indicate their own students. Apology is tendered to those omitted. Publications included in the bibliography are from two sources: The author and The Mathematical Reviews. An astounding wealth of mathematical capability and contributions, stemming from Moore is displayed. Of much concern in this writing has been the awareness of demands made by Robert Lee Moore on himself and others regarding exactness and integrity of statements. In no sense is it suggested that the statements included are of the sense of exactness which would have been demanded by 3

Robert Lee Moore of himself, else the activity yet would remain unﬁnished. However, much, much eﬀort has been made to assure the accuracy of statements included. Repetition does occur from one chapter to another. Some of this is purposeful, anticipating that “assigned readings” might extract those sections of most interest to the audience. Conviction has grown, throughout the preparation of this material, that not enough recognition has been given George Bruce Halsted — for his own teaching and mathematics. He must have carried with him contagious inspiration. Throughout, much dependence on others is admitted and appreciated. Indeed, the undertaking would never have occurred without the assistance of friends and former students of Robert Lee Moore.

D. R. Traylor University of Houston 1972

Part I

Chapter 1

The Early Years

By the fall of 1882, the opening of the Civil War was barely more than twenty years ago and the close of that terrible war between the states was barely more than fifteen years ago. The Alamo had fallen only 46 years earlier and Texas had been a state for only 37 years. Sam Bass, the leg- endary “Robin Hood” train robber had been shot down in the dusty street of Round Rock, Texas only five years ago. Carrie Nation was still operating a hotel in Richmond, Texas and yet was to make her way to Kansas and prohibition fame. The University of Texas had not yet experienced its first class. Dallas, a leading Texas city was indeed still only a frontier town. Open saloons were still in business in Dallas in 1882. Buttermilk had become a very popular drink and was available in the saloons. During the decade from 1880-1890, the population of Dallas had more than doubled and by 1900, Dallas County exceeded any other county in the state in population. In the twenty year period from 1880 to 1900, the city built its first reservoir for fresh water. Communications were improved when the Dallas Telephone Exchange was established in June 1881. It would accommodate 1200 phones and about 260 hand operated phones were immediately put to use. In 1882 the first electric light plant was put into operation. Railroad facilities serving the city were greatly expanded and electric street cars appeared on Dallas streets, although transportation was still horse dominated at that time. The first paving of any Dallas street appeared in 1882 when two main streets, Elm and Main, were paved with bois d’arc blocks. The first traffic arrest was made in early 1886. The charge was driving too fast down Main Street and the fine was $1.00. The prices of goods of that period are suggested by Sanger Brothers offering men’s worsted suits for $10.50 each. Whiskey could be purchased for $1.00 per gallon. The year 1882 was unusually dry since only 25,900 bales of cotton were received in

7 8 CHAPTER 1. THE EARLY YEARS

Dallas, down from 47,600 of the year before. In 1885 the Dallas Morning News began publication, as did the Herald, the forerunner of the Dallas Daily Times Herald. In 1886 patrons at the Dallas Opera House thrilled to James O’Neill in “The Count of Monte Cristo” and in 1887 paid $15.00 each for seats when Edwin Booth reenacted the role of Hamlet. The Dallas City Hospital was opened and the first public schools were opened to the children of Dallas, though there were many private schools and academies. Such was the setting into which Robert Lee Moore was born. He was the fifth of six children born to Charles J. Moore and Louisa Ann (Moore) Moore. He was to be raised in a home which honored the Southern attitudes, stressing gentlemanliness, honor, integrity, and independence. Charles J. Moore was an independent, strong-willed man who had come from New England, near Hartford, Connecticut. He had followed a brother to Kentucky before the beginning of the Civil War, leaving his northern home and two other brothers behind. He was to conceal that birthplace for many years, having totally joined the southern cause. Charles Moore volunteered for service in the armies of the Confederacy, serving in the Orphan Brigade, Company A, Second Kentucky Regiment. The brigade had been organized on October 28, 1861 at Bowling Green, Kentucky and fought as a unit for the first time at Shiloh. During 1862 the brigade found itself in Vicksburg and suffered more from malaria than from the Union fleet which was attacking Vicksburg. After six weeks at Vicksburg the number of men fit for duty in the brigade had dropped from 1822 to 548. 1 The brigade left Vicksburg only to return in 1863 in an effort to relieve the garrison there which was under siege from the Union army under command of U. S. Grant. Failing in this effort, the brigade was sent to the Army of Tennessee, forming its rear guard as the Army of Tennessee was driven off Missionary Ridge at the Battle of Chattanooga. By May 1864 the brigade found itself still with the Army of Tennessee under command of Johnston, being slowly forced back by Sherman’s army, through Rocky Face Ridge, Resaca, Calhoun, Allatoona, Kennesaw, Pine Mountain, Mari- etta, until Johnston was replaced by Hood. The series of battles for Atlanta began and, upon the breaking of the Confederate army there, the brigade was mounted as cavalry and remained on the flank of Sherman’s army, doing what damage it could. The Orphan Brigade and the calibre of its men were anything but ordinary, as suggested by the following comments which were included in a book written by a survivor of that brigade:

Nathaniel Southgate Shaler, the noted Harvard geologist and anthropologist, was a native Kentuckian whose sympathies were

1Johnny Green of the Orphan Brigade, edited by A. D. Kirwan, University of Ken- tucky Press, 1956, p. XV. 9

with the Union during the war. Years later he was making a study which called for the observation of some group of about 5 000 soldiers who were of homogeneous ethnic background and whose ancestors had been for several generations on this conti- nent. After considerable research the only group he could find which met these qualifications was the Kentucky Brigade and he made an intensive study of its history. “From the beginning,” he said, “it proved as trustworthy a body of infantry as ever marched or stood in the line of battle. “He pointed out that in the hundred days of the Atlanta campaign it was almost continuously in action or on the march. More than eleven hundred strong at the beginning of that campaign, the Brigade suffered 1,860 fatal or hospital wounds. “At the end of this time there were less than fifty men who had not been wounded during the hundred days. ” “A search into the history of warlike exploits,” Shaler wrote, “has failed to show me any endurance to the worst trials of war surpassing this . . . The men of this campaign were at each stage of their retreat going further from their fire- sides. It is easy for men to bear great trials under circumstances of victory. Soldiers of ordinary goodness will stand several de- feats; but to endure the despair which such adverse conditions bring for a hundred days demands a moral and physical patience which, so far as I have learned has never been excelled in any other army. ”2 After the war Charles J. Moore met and married Louisa Ann Moore and migrated toward the West, finally settling in Dallas, Texas. He operated a hardware store and feed company in a central location in downtown Dallas. It was just off the town square and faced the site of the courthouse. In fact, by 1882, a fourth courthouse was being constructed almost directly across the street from the Moore feed and hardware store. (Years later, shots would ring out over the site of the old Moore hardware and feed store as John F. Kennedy, President of the United States, was assassinated on what was, by then, a grassy knoll in front of that same courthouse building. )The courthouse was an imposing structure with its stones topped by a magnificent square tower into which was built a huge clock whose four faces could be seen from every direction. So it was that Charles J. Moore had forsaken his home, friends, and relatives in the North, made his way south to fight on the side of a losing cause. He felt so strongly about the issues surrounding the war between the states that he looked upon the accident of his birth, having been in the

2Nature and Man in America, Nathaniel Southgate Shaler, New York, 1897, p. 277. 10 CHAPTER 1. THE EARLY YEARS

North, as a misfortune he would not expose. Even his children were not informed of his Connecticut birthplace until they were grown and learned of it almost by accident. He was happy to let people assume that he was from Kentucky. As is often the case in times of war, people of native leadership talents or other capabilities have little opportunity to allow those talents to flourish and prosper. It well could be that Charles J. Moore was one of these. In any case, his life in Texas became one of that of a merchant, operating a hardware and feed store. Schooling in the late 1800’s in Dallas, Texas was not well organized. The public system was not effective. As a result, many of the parents, who possessed the capability, would send their children to private schools. Often these would be operated by a person who would be called a headmaster or a principal. Quite often one person, or perhaps, a man and his wife, would teach everything that would be taught in such a school. It was to a man named Waldemar Malcolmson that Robert Lee Moore was sent. Each of his elder brothers and a sister had been sent to a private school and it was to this one that he came. Waldemar Malcolmson was an unusual teacher. One occurrence that is recorded dealt with a parent, Mr. W. H. Flippen, who brought his two sons to be enrolled in Malcolmson’s school. As the parent was about to depart, Malcolmson called after him and said, “Oh, Mr. Flippen, how do you want the earth taught, – flat or round? I can teach it both ways. ”3 The choice of Waldemar Malcolmson’s school, for young Robert Lee Moore, seemed to be a natural one. Two older brothers had also attended that school and, finding it satisfactory, his parents chose that he also should study with Malcolmson. He began school when he was eight and studied with Malcolmson until he was barely 15. His studies included such subjects as Spanish and shorthand, as well as other fundamental subjects. After learning shorthand, he habitually wrote notes to himself in the margins of his books. There had been established a public school district in Dallas in 1877, but the first school tax was not imposed until 1882. Public schools simply were not developed well enough to serve adequately the people of Dallas. Waldear Malcolmson’s school was only one of several private schools which had been organized to respond to the educational gap caused by the absence of public schools. Malcolmson’s school was known by several names and met, across the years, in at least three locations. Known as Central Academy, its address was Harwood and Live Oak Streets, as Dallas Lyceum its address was on San Jacinto Street, and while called simply Professor

3The Lusty Texans of Dallas, John William Rogers, E. P. Dutton and Co., Inc., New York, 1960 p. 302. 11

Malcolmson’s school, it was located at 540 Pearl Street. Beginners studied addition, subtraction, reading and writing. Older students studied geography, higher arithmetic, spelling, composition, history, and penmanship. The spelling bee method was apparently used to teach a number of subjects other than spelling, including geography and pronunciation. At times a special instructor taught French classes. Anne Atkins, who attended Malcolmson’s school in the 1880’s gives the following description of Malcolmson: He was a rare type and a real scholar, with an odd approach to many things and unorthodox to a degree. He believed that Bacon wrote Shakespeare and was indifferent as to truths in the world around us. But when it came to mathematics, French or Latin, he ranked with the best and prepared another young playmate so well that he entered the University of Texas with the highest grades and became one of the most distinguished members of the State Bar, reading French and Latin with the greatest ease to the end of his life. The professor had an odd way of impressing facts upon us and one thing I have never forgotten–the names of the presidents of the United States up to Cleveland’s first administration; this was through learning the final initial of each. From the beginning he gave me, I have gone to a wide development of the French language and literature. 4 The Thomas collection of the Dallas Historical Society contains a monthly grade report from the Dallas Lyceum, 540 Pearl Street, issued by Professor Malcolmson in 1882. The report includes the following information: The second session of the institution will commence on the first Monday in September and continue, without intermission, for ten months. The method of instruction adopted is new, concise and lucid, not burdening the minds of the young scholars with useless matter, but omitting nothing necessary to a thorough, practical course of instruction. The Lyceum aims to be second to no institution of learning in Dallas; and a student will be able to receive instruction in all the branches taught at schools of the highest grade without leaving the building. Male and female students will sit in separate apartments, meeting only to recite.

4Information received from the Dallas Historical Society, February 1971. 12 CHAPTER 1. THE EARLY YEARS

Primary department subjects listed included orthography, reading and writing, arithmetic, grammar, geography, and U. S. history. Intermedi- ate students were offered algebra, advanced geometry, geography, modern history, natural philosophy, rhetoric, composition, and Latin. Courses in the senior department included trigonometry, physiology and hygiene, elementary chemistry and astronomy, ancient history, mental philosophy, and Latin. At extra cost students were offered classes in German, Spanish, French, mathematics, drawing, bookkeeping, and painting. Malcolmson and his wife may have been a bit ahead of their time in some respects. She had acquired the habit of smoking and came to an unfortunate end by setting her bed on fire. 5 Latin was not taught by Malcolmson and neither was calculus, after Robert Lee Moore began his studies with him. Moore needed Latin for entrance into the newly established University of Texas and, since he wished to study mathematics, he borrowed a calculus book from his teacher,6 with which to begin his study of calculus. Soon, though, he completely lost patience with its imprecise language and description. So he wrote to the University of Texas requesting a copy of the calculus book used there. He was to recall, in the twilight of his career, the pleasure with which he studied the new book, and then stated that, “I don’t think as highly of it now as I did then. ” His method of studying calculus, using the book sent to him, was natural for Robert Lee. He would read a statement of a theorem, but would intentionally cover the portion of the page which gave the proof of the theorem. Thus, he would attack the theorem on his own. If, after what he felt a reasonably long time was spent without success in proving the theorem, then he would uncover the first line of the proof, read that, and then try to prove the theorem without further assistance. If this failed, he would uncover the next line and continue on in that fashion until he had obtained a proof. It wasn’t a pleasant experience for him to uncover even a part of the proof and, in those instances that he uncovered much of the proof, he felt as though he had failed. However, he needed to have completed his calculus by September so that he could be received at the University of Texas with an understanding of calculus, so he proceeded through the calculus in that fashion. This same attitude about reading or learning of the work of other people was held consistently by Robert Lee Moore throughout his life. Sometimes he would attend a mathematics lecture. His custom was to seldom attend

5The Lusty Texans of Dallas, John Willlam Rogers, E. P. Dutton and Co., Inc., New York, 1960, p. 302. 6Almost 70 years later, while being interviewed in a film about his teaching methods, he was to recall that he still had that calculus book. 13 such, but in those rare circumstances in which he decided to, he would invariably ignore the speaker and contemplate problems of his own. In one instance, he did notice the theorem the speaker proposed to prove. He found it interesting and began to work it out for himself. At the end of the talk, when the speaker was responding to questions, Moore spoke up, stating that he, too, had a proof of that theorem. Upon going to the board and indicating his proof to the speaker, he learned, only then, that his proof was the same as that just presented by the lecturer. Another time, after he had received high office in the American Mathe- matical Society, he wandered into a classroom, while attending a meeting, to find two mathematicians, Lefschetz and Weiner, in deep discussion at the board. Moore said, ”What are you doing?” They replied, ”Well, it’s a mathematics meeting, isn’t it? We’re discussing mathematics.” Moore’s response came, ”Well, it’s a mathematics meeting, but it seems to me that’s the last thing you ought to be talking about!” When he related this conversation later to a class of his, he added, ”They didn’t understand what I meant.” Sometime prior to Robert Lee Moore’s fifteenth birthday, he decided to enroll at the University of Texas. It was a result of that decision that he dropped out of Malcolmson’s school, to study independently those subjects which he needed to enter the University of Texas. He enrolled there for the first time a few weeks before his sixteenth birthday, in 1898. The University of Texas was barely older than Robert Lee Moore. Al- though it had been authorized many years before, the first meeting of the University Regents was held November 16, 1881, almost exactly one year before Robert Lee Moore’s birth. The Regents authorized the organization of the academic and law departments. The university was formally opened with public inaugural exercises on September 15, 1883. The University of Texas began as a co-educational institution, with the statute under which it was organized stating that, ”it shall be open to all persons in the State who may wish to avail themselves of its advantages, and to male and female on equal terms.”7 The annual attendance of students during the first seven years of the operation of the university is recorded as

7History of the University of Texas, J. J. Lane (Henry Hutchings State Printer, Austin, 1891) p. 249. 14 CHAPTER 1. THE EARLY YEARS

Sessions Academic Law Department Total 1883-84 166 52 218 1884-85 151 55 206 1885-86 138 60 198 1886-87 170 73 243 1887-88 176 73 249 1888-89 187 91 278 1889-90 230 78 308

On November 17, 1882, again almost on Robert Lee Moore’s birthday, the cornerstone of the Main Building was laid. The building was the first on the campus. Only the west wing had been constructed by 1883, when the first students enrolled and by 1889 the grand central section was barely completed. Construction of the east wing was delayed for lack of resources. With the coming of the fall of 1883 came the opening of the University of Texas. Eight chairs had been filled and those eight men, along with five assistant faculty, were the faculty of the University of Texas in 1883. They were

Professor J. W. Mallet, A.M., M.D., LL.D., Ph.D., School of Chemistry and School of Physics. Professor Mallet was a native of England, but had become a citizen of this country prior to the Civil War. He served in the Confederate army with rank of colonel. Before coming to the University of Texas, he was on the faculties of the universities of State Geological Seminary of Alabama, Alabama,and Virginia. Professor William LeRoy Broun, A.M., LL.D., School of Mathematics. Professor Broun was a native of Virginia and served in the Confederate Army as a colonel in the ordnance department. He was a member of the faculty of the Univer- sity of Georgia, President of the State College, a member of the faculty of Vanderbilt, and President of the Agricultural and Mechanical College of Alabama before accepting a post at the University of Texas. Professor Milton W. Humphreys, A.M., LL.D., Ph.D., School of Ancient Languages. Professor Humphrey was born in West Virginia. Before coming to the University of Texas, he was on the faculties of Washington and Lee and Vanderbilt. He served for one year as President of the American Philological Associa- tion. Professor Leslie Waggener, A.M., LL.D., School of English Language, History and Literature. Professor Waggener was a 15

native of Kentucky. He received an undergraduate degree at Harvard and the LL.I). from Georgetown College. He was a member of the faculty of Bethel College, Kentucky and also serve as President of that institution before joining the faculty of the University of Texas. Professor R. L. Dabney, A.M., D.D., LL.D., School of Men- tal and Moral Philosophy and Political Science. Professor Dab- ney was born in Virginia, and served in the Confederate army as General Stonewall Jackson’s chief of staﬀ. He was educated in Virginia and ordained a minister in the Presbyterian church. He was on the faculty of the University of Virginia. Professor H. Tallichet, B.L.D., School of Modern Languages. Professor Tallichet was born in France and studied in various schools in Europe. He taught in this country in Baltimore, Wilmington, Ashville, Charleston, and the University of the South, Sewanee, Tennessee. Professor Oran M. Roberts, A.M., LL.D., and Professor Robert S. Gould, A.M. Both Professor Roberts and Professor Gould received degrees from the University of Alabama and served as Chief Justice of the Supreme Court of Texas. Profes- sor Roberts also served as Governor of the state of Texas8 Until 1897 students could be admitted to the Academic De- partment of the University of Texas upon passing an examination in English. In 1897, to that requirement were added subjects of history and mathematics. This ended to cause university students to be drawn from the better high schools and academies. The long session was broken into three terms, called respectively Fall, Winter, and Spring Tens. The long session began on the fourth Wednesday in September and closed on the third Wednesday in June. The Fall Term closed just a few days before Christmas. The Winter Term opened shortly after the beginning of the year and ended on the third Saturday in March, with the Spring Term beginning on the ﬁrst Monday following the third Saturday in March. The catalogue requirements had been strengthened by 1897, prescribing all studies for the freshman year and requiring each of those to be completed before the student could take up other studies. After the freshman year, one course was prescribed in each of the following subjects: Political Science, History, Philosophy, Nat-

8History of the University of Texas, J. J. Lane (Henry Hutchings State Printer, Austin, 1891) p. 249. 16 CHAPTER 1. THE EARLY YEARS

ural Science, and English and the remainder of the course was entirely elective.”9

Thoroughness of instruction was of concern in 1897. As an example, the School of Mathematics had established a new instructorship, thus enabling the teaching force of that school to divide the freshman class into six sections instead of two. To those six sections the entire time of the two instructors of mathematics was devoted. The smaller sections enable the student to get more individual instruction from his professor; his enthusiasm and interest is quickened and aroused, and a desire created to pursue the subject in its higher branches. The percentage of failures has largely decreased.”10 The total enrollment of academic students by 1897 was 408. The library held approximately 2640 volumes and the university authorities claimed earnest desire to reduce to the lowest possible point the expense of education. The spirit of the institution is favorable in economy in dress and in living. There is no prescribed uniform, nor is there such devotion to fashion among the students as makes it unpleasant for any one to follow the utmost economy during this University life. Some students do their own cooking and house work, and are thus enabled to live at an expense not exceeding $5 a month. They serve as waiters in boarding houses, or do other work in private families, which relieves them of expense of board. Regular board, with furnished room, can be obtained near the University, at prices varying from $12.50 to $20 a month. A large number of students pay the former price. In University Hall board, furnished room, lights and fuel may be obtained for $15 a month. Two large student clubs have, during the present session, further reduced the price of board and lodging. The Thomas Arnold Club have lived at an average expense of $11.25 monthly for each member.11 Many of the students supported themselves by doing work in private families, milking cows, making ﬁres, cooking, tending the horse; others

9The University Record (The University of Texas, Austin, Texas, 1898) Vol. I, No. 1, December 1898, p. 30. 10ibid. pp. 32-33. 11The University Record (The University of Texas, Austin, Texas, 1899) Vol. I, No. 2, April 1899, p. 93. 17 waited on the tables in boarding houses or attended to the rooms; others taught, acted as clerks, stenographers, typists, accountants, or surveyors. One student, while proceeding with his graduate studies stated: As the notion that those of limited means are unable to attend the University of Texas is more or less prevalent over the State, I desire to combat the idea, having as my grounds for so doing an experience of ﬁve years within her walls. By assistance of a friend I secured a pleasant home in Austin, where I paid my board by working on the lawn and in the house. I entered the University in February, 1894, and continued to work in a private family for my board until in January, 1898, when I secured a better position in the University. My expenses in the University have been about as follows:

From February 1, 1894 to June 20, 1894 $ 7.85 From September 1894 to June 1895 $70.00 From September 1895 to June 1896 $60.00 From September 1896 to June 1897 $25.00

Total for four years $227 85

But during the session of 1896-1897 I earned $40 by acting as janitor of the literary societies. Also during the session of 1897- 1898 I earned $64 by doing private teaching. Subtracting this $104 from the total amount above given, leaves a total outlay for four years of $123.85. I need only to add that during the four years in question I never failed to make the studies I undertook, that is, ﬁve courses a session.12 The academic standards which Robert Lee Moore found confronting him in 1898 are suggested by: Admission on Examination Candidates for admission to the B. A. course will stand entrance examinations in English, History, Mathematics, Latin or Greek. Candidates for admission to the B. Lit. course will stand entrance examinations in English, History, Mathematics, and in Latin or Greek. Candidates for admission to the B. A. or the B. Lit. courses who cannot take the entrance examination 12ibid., pp. 94-95. 18 CHAPTER 1. THE EARLY YEARS

in Greek, may take Greek A in the University (see School of Greek). Candidates for admission to the B. S. course will stand entrance examinations in English, History, and Mathematics; and, beginning in 1901, also in a natural science or in a modern language. Applicants for admission on examination who do not wish to become candidates for a degree, will stand entrance examinations in English, History, and Mathematics. 1. In English candidates will be examined upon their knowledge of the elements of English Grammar and English Composition, and especially upon their ability to write simple paragraphs of idiomatic English properly spelled and punctuated. The subjects for such paragraphs will be taken from the books named below, whose subject matter the candidates must be familiar with: in 1899, the Sir Roger deCoverly Papers in Addison’s The Spectator, Dryden’s Palamon and Arcite, Cooper’s The Last of the Mochicans, Burke’s Speech on Conciliation with America; in 1900, Scott’s Ivanhoe, Tennyson’s The Princess, Macaulay’s Essays on Milton and Addison, Dequincey’s Flight of a Tartar Tribe. As an important part of his work of preparation the student should be encouraged to read widely in good English literature, and should be given constant practice in writing idiomatic English in connection with his reading. 2. In History the examination will cover the outlines of universal history. The amount of knowledge required to pass in this subject will be indicated by Myer’s Outlines of Gen- eral History. 3. In Mathematics the examination will embrace arithmetic, including the metric system, algebra through equations of the second degree, and plane geometry. 4. In Latin candidates will be examined in grammar, with special stress upon inﬂections and the syntax of the simple sentence; the translation of elementary English prose into Latin; in Viri Romae, any books of Caesar, the four lives of Nepos that bear upon Roman history (Hamilcar, Hannibal, Cato, Atticus), any four orations of Cicero, and the ﬁrst book of Virgil’s Aeneid with the scansion of the dactylic hexameter. 5. In Greek candidates will be examined in grammar, on in- 19

ﬂections and syntax; in any three books of Xenophon’s Anabasis; in the translation of easy Greek at sight; in the translation of elementary English prose into Greek. Knowl- edge of accent is required.13 The courses and degree requirements were described thusly: Courses of Instruction The courses oﬀered in the Department of Literature, Sci- ence, and Arts are either one-third, two-thirds, or full courses, according to the estimated amount of work in each. A full course occupies three hours a week throughout the session; a one-third course one hour a week throughout the session or three hours a week for one term; and a two-thirds course two hours a week throughout the session or three hours a week for two terms. Twenty full courses, or their equivalent, are required for every baccalaureate degree. Courses are distributed in most branches of study into three groups: those designated for undergraduates; those open to advanced undergraduates and to graduates; and those open to graduates.

Requirements for the Baccalaureate Degrees

To attain any one of the baccalaureate degrees the candidate must satisfactorily complete twenty full courses or their equivalent, and must produce a creditable essay or address at graduation. Some of these courses are prescribed; others are elective.

For Bachelor of Science

Freshman Year English 1, 1 course. • Mathematics 1, 1 1/3 courses. • A modern language (French, German, or Spanish), 1 1/3 • courses, or History, l course. A modern language (French, German, or Spanish), 1 l/3 • courses, or a science, (Biology, 1 course, or Chemistry, 1 1/3 courses). Physiology and Hygiene, l/3 course. • 13ibid., pp. 102-103. 20 CHAPTER 1. THE EARLY YEARS

Physical Culture. • Sophomore Year English 2, l course. • A modern language (Scientific French or Scientific Ger- • man) l course. A science, l course. • Elective Courses The remainder of the twenty full courses may be selected the candidate, subject to the following conditions: (a) Three full courses must be completed in some one subject; for B. S. candidates these courses must be in a natural science. (b) None of the courses prescribed above can be taken in the Senior year, excepting Biology and Chemistry. (c) Every candidate for a degree is required to take one full course in each of these subjects: Political Science (El- ements of Political Economy and Government), History, Philosophy, and Natural Science. (d) Students must select their studies in conference with the Advisory Committee of the Faculty; and no student will be al]owed to register for courses of study not approved by this Committee. (e) In addition to the completion of twenty full courses every candidate for graduation is required to prepare and hand to the President by April l of the year of his intended graduation a creditable essay or address, whose title must be submitted to the President for approval on or before the first day of December. From the essays or addresses accepted, the Faculty will select one or more for delivery in public on Commencement Requirements for Master of Science Degree For the degree of Master of Science the requirements are as follows: 1. A prior degree of Bachelor of Science of the University of Texas, or of another institution; provided, that in the latter case the Faculty must be satisfied that the courses pursued are equivalent to those required by this University. 21

2. The equivalent of five full courses of graduate instruction satisfactorily completed; three-fifths of the work to be pros- ecuted in one school, such time as the instructor in charge may approve being devoted to the preparation of a thesis; the remaining two-fifths to be selected outside that school. 3. At least one year of residence at the University; a residence period longer than one year in case outside duties unduly encroach upon the time necessary to the satisfactory completion of the required work. 4. The approval of the course of study by the Advisory Committee- on Graduate Degrees, and the approval of the thesis by the professor in charge of it and by the committee.

Thesis

Every candidate for a master’s degree must communicate to the President the title of his proposed thesis on or before the first Monday in March of the year in which he intends to present himself for final examination, and must hand to the President a fair copy of his thesis on or before the first Monday in May. The thesis with a certificate of approval will be deposited in the Library for public inspection. A successful candidate for a master’s degree is allowed to print his thesis as one accepted for the degree, with the signed certificate of approval; and either a printed or written copy of the thesis and the signed certificate must be permanently deposited in the Library and remain open to public inspection. The title of his thesis will be in the Commencement programme and in the next following annual Catalogue.

University Duties

No student will be allowed to register for more than ﬁve and one- third courses of study, except on petition approved by the Advi- sory Committee. A course is three recitations a week throughout the year, and thus the average number of recitations a week for each student is sixteen. A recitation lasts one hour. No student under twenty-one years of age is allowed to take less than four courses, or twelve recitations a week. Special students, being over twenty-one years of age, may do work in only one or two subjects, and may take fewer recitations a week than is permitted a regular student. 22 CHAPTER 1. THE EARLY YEARS

Uniform and punctual attendance upon all the exercises of the University is strictly required. Students absent from any exercises of the University at which they are due must present their excuses to the President not later than the day after their return to their classes. A student leaving the city during the session of the University must ﬁle an application for a leave of absence, and no student is permitted to withdraw permanently until he has received a certiﬁcate of honorable dismissal from the President. The University of Texas considers its students men and women, and consequently they are under no other restriction than those imposed by good society. The young women have the advantage of the presence of Mrs. Helen M. Kirby, who has been appointed by the Board of Regents Lady Assistant, and an opportunity for daily conference with her. Mrs. Kirby gives her entire time to looking after their health and comfort. A private study room is provided for them. The general Library is open to all students, and here order and quiet are necessarily maintained. The ”Honor System” prevails on examinations.

Courses oﬀered in the School of Pure Mathematics included:

1. Spherics, Solid Geometry, Algebra, Plane and Spheri- cal Trigonometry, with Applications to Surveying and Naviga- tion. 2. Conic Sections, Analytical Geometry. 3. Calculus for Physics and Engineering. 4. Differential and Integral Calcu- lus. 5. Integral Calculus, Differential Calculus, and Differential Equations, for Physics, Engineering, and Economics. 6. His- tory of Elementary Mathematics. 7. Advanced Integral Cal- culus: Definite Integrals, Differential Equations, Functions of a Complex Variable. 8. Modern Geometry, Metric Geometry, Recent Geometry. 9. Geometry of Position. 10. Theory of Equations, Theory of Functions. 11. History of Mathematics. 12. Non-Euclidean Geometry. 13. Hypergeometric Functions. 14. Algebra of Logic.14

With the opening of the session 1884-1885, one notable change had occurred. Professor William LeRoy Broun no longer held the chair in mathematics. Replacing him on the faculty was George Bruce Halsted Professor of Pure and Applied Mathematics. Halsted had earned his M. A. from Princeton and had received his Ph.D. from Johns Hopkins University. It

14ibid., pp. 105-109, 111. 23 was to Halsted that Moore would come in 1898 and it was about Halsted that Moore would later say, “There is no other person I would have wished to study under; that is, if someone should ask me if I wouldn’t rather have studied under Professor X, and they named any other person, then I would say, ’No!’ If someone were to ask me if I wouldn’t have preferred to begin my university studies under E. H. Moore, I would have said, ’No!’ There is no one I would have preferred over Halsted!”15 One description of Halsted and his teaching of R. L. Moore was given by Burton Jones:

George Bruce Halsted came to the University of Texas as Professor and Head of the Department of Mathematics in 1884. He was a man of character, strong opinions, and he is best remembered for his translation of Poincare’s popular essays on Science and Hypothesis. Halsted’s mathematical interests were also broad but mainly centered on geometry. Combining this interest with his interest in teaching he wrote several text books on geometry. One of the most exciting topics of the times was the foundations of geometry. Non-Euclidean geometry had been discovered and Hilbert was formulating his famous axioms. Hal- sted was abreast of these developments and incorporating them as quickly as possible into his teaching. So it was to this man that Robert Lee Moore came in 1898. Moore was a strong willed, to some extent self educated, young man of almost 16. Quick of mind, already with a driving interest in and dedication to mathematics, he was placed by Halsted in calculus. After a short period of time when it became evident that calculus was not suﬃciently challenging, Halsted transferred Moore to his course in projective geometry. Thus in his freshman year he was already in competition with juniors and seniors.16

Halsted was not a person who many would call a good teacher. His style of teaching was not that to which the typical student had been accustomed. Halsted lectured, or talked, often in class but seldom about mathematics. He would speak of his travels, his experiences, and his attitudes without much apparent reservation. His treatment of mathematics was done by way of calling on students to explain passages in the textbook they were using. It was usual that some assignment had been made in the text at the preceding class and Halsted would commence by asking some person

15do we need stuff here? 16Proceedings, Emory Topology Conference (Emory University, Atlanta, Georgia, 1970) edited by J. W. Rogers, Jr., p. 10. 24 CHAPTER 1. THE EARLY YEARS in the class to explain some statement from that section which had been assigned. If, for instance, a phrase had begun in the text with, “It is clear that. . . ” or “It follows from the above that. . . ” or some other statement suggesting that certain details were omitted, Halsted would start by saying, “Mr. BLANK, will you explain to us how that follows?” If no answer were made, Halsted would proceed on o another member of the class, asking, “Mr BLANK will you explain how that follows?” After inquiring of several other 1 students, without result, he would then be likely to say, “All right, Mr. Moore, how does it follow?”17 However, such a description of Halsted and his classroom manner does not offer the full flavor of the man. Coming to Texas, joining a faculty which contained among its few members a distinguished old soldier, Dabney, who was Stonewall Jackson’s chief of staff during the Civil War, Halsted held his own with all of them in terms of individualism and strength of personality. T. U. Taylor, a member of the faculty at the University of Texas until 1938, recalled Halsted:

George Bruce Halsted was a graduate of Johns Hopkins Uni- versity and was an investigator that was ever on the quest. While some have said that he pursued not the standard lines but morphological forms of the subject; still in fairness he must have credit for his great industry and his faculty of inspiring young men to do research in mathematical ﬁelds. He was a searcher in many languages, and if he got the scent, he was a regular blood hound on the track of new game. America owes it to George Bruce Halsted for its knowledge of Non-Euclidean Mathematics, and it was he that republished the monographs of Lobachevski and Bolyai. He was never a detail man for the exactness of a second or fraction thereof but an investigator for those forms of mathematics that the American Mathematicians had neglected or ignored. His services at the University of Texas were terminated about 1900 or 1901 on account of misunderstandings. He was too free in his criticisms of the University authorities and matters became acute in the month of December one year, and the Board met and severed his connections with the University suddenly. Dr. Halsted has published many books: Mensuration, Plane and Solid Geometry; Lobachevski, Bolyai and many monographs and articles in magazines. He was a prodigious writer and was

17Personal conversation with R. L. Moore, Austin, Texas, January 26, 1972. 25

a constant attendant at the National meetings of the scientific bodies.18 One account of Halsted’s attendance at a meeting of the American Mathematical Society was recalled by Professor F. Morley: The meetings of the Society were friendly, optimistic, and even jovial. The seniors, such as G. W. Hill and H. A. Newton of Yale, were of course dignified; and so were the Harvard men. But it was a new note in scientific meetings to encounter, for instance, G. B. Halsted, who said to me when we met: Come down to Texas, and we will shoot Mexicans.19 In no sense was George Bruce Halsted subservient to others, who perhaps enjoyed greater mathematical reputation than he. For instance, at the first summer meeting of the American Mathematical Society, Halsted had this to say about E. H. Moore: The paper by Professor Eliakim H. Moore of the University of Chicago was a piece of padding. Of course in an elementary class on the theory of functions it would be a good exercise to have each of the students write a half dozen such papers, and in a meeting lacking material such padding, like cotton in a tooth, holds the opening until the gold comes. The same remarks are true of the paper read by Professor Moore in Section A of the American Association.20 At the time Halsted made that comment, one of his proteges, Dickson, was just beginning his graduate study at Chicago and R. L. Moore was several years away from going there. In the language of that period, it is clear that Halsted did not “toady” after anyone! Another description of Halsted suggests the unusual qualities of the man: Of all the rare and odd professors that have been on the Faculty of the University of Texas, I think George Bruce Halsted will rank number one. He came to the University in the fall of 1884 during the University’s second year and for about sixteen years his sayings

18Fifty Years on Forty Acres, T. U. Taylor (Alec Book Company, Austin, Texas, 1938) p. 87. 19American Mathematical Society Semicentennial Publications (American Mathemat- ical Society, Providence, Rhode Island) Vol. I, History, p. 10. 20I’m not sure this location is correct! American Mathematical Monthly, Vol. I, No. 8, 1894, pp. 287-288. The Early Years, continued. 26 CHAPTER 1. THE EARLY YEARS

and doings in the classroom and in public lectures were the talk of the campus and the town. In the early days he made a trip to Mexico and he conceived the idea that he had discovered a plant, the juice of which would revive old age and prolong a person’s life. He brought some of the plant home, boiled it down to a syrupy mixture, brought a bottle to class and had each student come up and take a dose of the elixir. Even one of the Deans of A. & M. College boasted to his dying day that he had taken a spoon of Halsted’s elixir. He had written a geometry and had hated the definition of a straight line as given in most books as the shortest distance between two points. He would always tell his class that a dog knows this or a dog knows that. One day he breezed into class and suddenly asked Mr. LacLane, Ry, “What does a dog know?” Mr. LacLane, Ry replied, “A dog knows that a straight line is the shortest distance between two points.” This was contrary to the Doctor’s sermon and he immediately proceeded to lecture about the knowledge of a dog and straight lines for an hour. He was rather caustic and made personal remarks to the students and commented on their replies. Some took offense and one day a personal encounter was prevented by the chairman of the faculty.21 Halsted came by his academic individualism honestly. He was a student of J. J. Sylvester, who had come to the United States from Great Britain. During his inaugural address before the Texas Academy of Science on October 12, 1894, Halsted related the following about Sylvester: . . . As Sylvester would not sign the thirty-nine articles of the Established Church, he was not allowed to take his degree, nor to stand for a fellowship to which his rank in the tripos entitled him. Sylvester always felt bitterly this religious disbarment. His denunciation of the narrowness, bigotry, and intense selfishness exhibited in these creed tests was a wonderful piece of oratory in his celebrated address at the Johns Hopkins University. No one who saw will ever forget the emotion and astonishment exhibited by James Russell Lowell while listening to this unexpected climax. Thus barred from Cambridge, he accepted a call to America from the University of Virginia. 21Fifty Years on Forty Acres, T. U. Taylor (Alec Book Company, Austin, Texas, 1938) p. 290. 27

The cause of his sudden abandonment of the University of Virginia is often related by the Rev. Dr. R. L. Dabney, as follows: In Sylveser’s class were a pair of brothers, stupid and excruciatingly pompous. When Sylvester pointed out one day the blunders made in a recitation by the younger of the pair, this individual felt his honor and family pride aggrieved, and sent word to Professor Sylvester that he must apologize or be chastised. Sylvester bought a sword-cane, which he was carrying when waylaid by the brothers, the younger armed with a heavy bludgeon. An intimate friend of Dr. Dabney’s happened to be ap- proaching at the moment of the encounter. The younger brother stepped up in front of Professor Sylvester and demanded an in- stant and humble apology. Almost immediately he struck at Sylvester, knocking off his hat, and then delivered with his heavy bludgeon a crushing blow directly upon Sylvester’s bare head. Sylvester drew his sword-cane and lunged straight at him, striking him just over the heart. With a despairing howl, the student fell back into his brother’s arms screaming out, “I am killed! He has killed me!” Sylvester was urged away from the spot by Dr. Dabney’s friend, and without even waiting to collect his books, he left for New York, and took ship back to England. Meantime a surgeon was summoned to the student, who was lividly pale, bathed in cold sweat, in complete collapse, seemingly dying, whispering his last prayers. The surgeon tore open his vest, cut open his shirt, and at once declared him not in the least injured. The fine point of the sword-cane had struck a rib fair, and caught against it, not penetrating. When assured that the wound was not much more than a mosquito-bite, the dying man arose, adjusted his shirt, buttoned his vest, and walked off, though still trembling from the nervous shock. Sylvester was made head professor of mathematics of the Royal Military Academy at Woolwich, a position which he held until the early period set by the English military laws for conferring the life-pension. He thus happened to be free to accept a position at the head of mathematics in the Johns Hopkins University at its organization.22

22Science Magazine, Vol I, No. 8, February 22, 1885, p. 205. 28 CHAPTER 1. THE EARLY YEARS

Taylor elaborated further on Halsted. He bought some Shetland ponies and drove over town at a rattling pace irrespective of traffic laws or right nd left side of the street. One day in the faculty meeting one of the professors happened to refer to the police court and ended with the remark that he had never before been before the police court. Immediately Dr. Halsted from his seat interjected, “I have the advantage f you. I have been there several times.” When he built his house on the lot where the University Methodist Church now stands, he erected it on brick columns high enough for a man to walk under without bumping his head. The students all said that he did this o the flood would not wash away his house if the Austin am ever broke. I cannot vouch for this statement but it as current and widespread on the campus, and it was typical. On one occasion he remarked in the faculty that an ordinary thesis ought not take a student over an hour to write. There is one outstanding thing that must be said to the credit of George Bruce Halsted: He inspired men to study and research and In this respect he made a genuine contribution to American scholarship in mathematics.23 Halsted and his Shetland ponies must have caused considerable attention in Austin. An ex-student, in giving comments before a gathering of ex-students in Dallas on October 21, 1899 recalled: . . . I walked down to Congress Avenue and while walking north, on the side of the Lobby, I saw a man of small stature and with a great stiff moustache that attracted my attention. He was standing astride the gutter through which the water was running, and was bent over picking up rotten apples and cutting the rotten parts from them with his knife. As soon as I could, I asked a man who that fellow was. He said, “I think it is Halsted.” “What,” I said, “George Bruce Halsted of the Uni- versity?” Looking again, he assured me it was. Such was my introduction to one whose books I had often seen and studied and whose name was so familiar to the scientists of every civilized nation. After he had picked up several of the apples, he walked to a Shetland pony hitched nearby, and with a smile lighting up his countenance, fed them to the pony. For a moment my

23Fifty Years on Forty Acres, T. U. Taylor (Alec Book Company, Austin, Texas, 1938) pp. 290-291. 29

respect for the man was lessened, but the more I thought of it, the more it weighed with me as evidence of his greatness, being that childish simplicity that so often characterized the truly great.24

Halsted had already experienced unusual success with his students before Robert Lee Moore enrolled at the University of Texas. One of Halsted’s ﬁrst successes came in the form of Milton Brockett Porter. Porter had come from Sherman, Texas, having been born there on November 22, 1869. He attended Austin College in Sherman and then was a student at the Uni- versity of Texas from 1889-1892. Leaving Texas, Porter received his M. A. from Harvard in 1895 and his Ph.D. from Harvard in 1897. He returned to Texas as Instructor in Pure Mathematics in 1897 and accepted a similar post at Yale in 1898. He was promoted to an Assistant Professorship there before long and held that position until he returned to the University of Texas as Head of the School of Pure Mathematics in 1903. A phrase describing Porter at the time of his return in 1903 was:

Gifted with much mathematical insight, and possessed of a large store of broad and accurate knowledge in many ﬁelds, his rare and admirable character and high ideals are sure to play a most important part in the development of the University.25

So at the time Robert Lee Moore entered the University of Texas, Porter had already been away to earn his doctorate, had returned, but would leave Texas for Yale before Moore graduated. Another student of Halsted’s of near Porter’s age was H. Y. Benedict who returned to Texas in 1899 as Instructor of Pure Mathematics.

Benedict had been born in Louisville, Kentucky in 1869 and came to Young County, Texas when he was eight years of age. Except for scattered schooling, he was prepared for college at home, by his mother. He entered the University of Texas in 1889, graduating at the head of his class of 1892. During 1891- 1892 he was a Fellow in Pure Mathematics, and in 1892-1893 a Tutor in Pure and Applied Mathematics at the University. He moved to the University of Virginia and during 1893-1895 he was an assistant in astronomy at the University of Virginia. He resigned that post to study at Harvard, graduating from there

24The University Record (The University of Texas, Austin, Texas, 1900) Vol. II, No. 1, January 1900, p. 35. 25The University Record (The University of Texas, Austin, Texas, 1903 Vol. V., No. 2, August 1903, p. 185. 30 CHAPTER 1. THE EARLY YEARS

with a Ph.D. “in mathematics, especially astronomy,” in 1898, having held two scholarships while at Harvard. He spent one year at Vanderbilt University, 1898-1899, having “entire charge of the department of mathematics at that institution.”26

Leonard Eugene Dickson was a third student of much success who had experienced early contact with Halsted.

He (Dickson) was from Cleburne, Texas where his father had been for years a successful merchant, having considerable holdings in real estate. Dickson entered the sixth grade of the Cleburne Public Schools at their opening session in 1883. He continued there ﬁve years and then had two years of academy instruction before entering the sophomore class of the Univer- sity of Texas, where he graduated in 1893 with ﬁrst honors of his class. During 1892-93, he was an assistant chemist in the geological survey of Texas. During 1893-94, he held the teaching fellowship in pure mathematics in the University of Texas, and completed the course for the degree of Master of Arts. He received, then resigned, an appointment to a Harvard fellowship, choosing instead to accept a senior fellowship in the University of Chicago. He held that post from 1894 to 1896. He gave elementary instruction in the college, graduating in 1896 with the Doctor of Philosophy. He then studied during 1896-97 with Sophus Lie at Leipsig and with Jordan, Appel, Picard, Her- mite, and Painleve at Paris. He was an assistant professor of mathematics at the University of California during 1897-1899, resigning that post to return to the University of Texas as an Associate Professor.27

Thus it was that by 1899 the faculty in the School of Mathematics at the University of Texas included: Professor George Bruce Halsted, Asso- ciate Professor L. E. Dickson, and Instructors H. Y. Benedict and T. M. PutmanPutnam was a promising young mathematician from the University of California who had studied with Dickson there, as well as having spent one summer studying at the University of Chicago. He came with Dickson from California to Texas. Porter had just left to accept a post at Yale. Halsted had gathered about him an able group of promising mathematicians: Porter who had begun work

26The University Record (The University of Texas, Austin, Texas, 1899) Vol. I, No. 2, April 1899, p. 223. 27The University Record (The University of Texas, Austin, Texas, Vol. I, No. 3., August 1899, p. 220. 31 with Halsted as an undergraduate and who made his way to Harvard for his Ph.D.; Benedict who followed almost exactly the same path, earning his degree in mathematical astronomy; and Dickson who began that route, only to be diverted to the newly established University of Chicago, earning his doctorate there before studying with the mathematical leaders of Europe. It was not exactly a sterile, impoverished mathematical community which Robert Lee Moore found himself in at the University of Texas. It was an exciting time of new mathematical discoveries and Moore could look about him and see others who had begun at Texas, gone away to pursue their Ph.D. elsewhere, notably Harvard and Chicago, and had returned with success achieved. An example of the mathematics oﬀering is given by a description issued in the University Record, Vol. I, No. 4, October 1899, pp. 359-360.

For Undergraduates 1. Spherics: Halsted’s Elementary Synthetic Geometry (third edition); Solid Geometry: Halsted’s Elements of Geom- etry (sixth edition); Algebra (Fisher & Schwatt); Plane and Spherical Trigonometry (Phillips & Strong), with applications to surveying and navigation (one and one-third courses; four hours weekly). Section I Professor Dickson Section II Instructor Benedict Section III Instructor Benedict Section IV Instructor Putnam Section V Instructor Putnam Section VI (For technical students) Professor Dickson 2. Conic Sections; Analytical Geometry - Puckle (full course; three hours weekly) Professor Halsted 3. Diﬀerential and Integral Calculus - Byerly (full course; three hours weekly) Professor Halsted 4. Introductory course in Analytic Geometry and Calculus, for technical students 1 full course; three hours weekly) Instructor Putnam 5. History of Elementary Mathematics (two-thirds course; two hours weekly). A History of Elementary Mathematics, with hints on Methods of Teaching, by Florian Cajori. Pro- fessor Halsted 6. Advanced Integral Calculus; Diﬀerential Equations (full course, three hours weekly) Professor Dickson 32 CHAPTER 1. THE EARLY YEARS

7. Modern Geometry; Metric Geometry; Recent Geometry (two-thirds course; two hours weekly) Halsted’s Mensura- tion (fourth edition); Halsted’s Synthetic Geometry (third edition). Professor Halsted 8. Geometry of Position (full course; three hours weekly). Halsted’s Pure Projective Geometry (second edition). Pro- fessor Halsted For Graduates 9. General Group Theory, Including Lie’s continuous groups with applications (full course; three hours weekly). Pro- fessor Dickson 10. Non-Euclidean Geometry (full course; three hours weekly). Halsted’8 Lobachevski (fourth edition); Halsted’s Bolyai (fourth edition). Professor Halsted 11. General Astronomy (full course; three hours weekly). Young’s General Astronomy. Instructor Benedict p 12. Spherical Astronomy and Orbit Theory (full course; three hours weekly). Instructor Benedict

By June 1900 Halsted oﬀered his description of the situation in the School of Pure Mathematics:

This year has been a bloom period for the School of Mathe- matics. In addition to the aid rendered by Mr. T. M. Putnam, of California, Professor Halsted ha had the extraordinarily able support of two of his own former students, Dr. Dickson and Dr. Benedict. For sixteen years, beginning with 1884, Dr. Halsted has given the work of the School a decidedly geometric character, believing that this, in its various ramifications, is the most re- munerative as it is the most charming part of all mathematics. But this year, in addition to the courses in Modern Synthetic Geometry, and in Recent Geometry of the Triangle and Circle (the Lemoine-Brocard Geometry), and in Geometry of Posi- tion (Projective Geometry), and in Non-Euclidean Geometry, the School has been strengthened and diversified by a course in Group Theory by Dr. Dickson and courses in Mathematical Astronomy by Dr. Benedict. Dr. Benedict has in preparation a work in Orbit Theory; and Dr. Dickson has become such an authority on Groups, that 33 the University of Chicago has offered him an Assistant Profes- sorship in which his advanced work is to be in that subject. His name already appears in the program of the Department of Mathematics and Astronomy for 1900-01. It is a source of great gratification to his former teacher and subsequent colleague here, that, while the University of Chicago emphasizes by special mention under Modern Mathematics, “synthetic geometry,” and under Introduction to the Higher Mathematics, “projective geometry,” yet with the advent of Dr. Dickson appears in its program the magic name “non-Euclidean geometrics.” Course 50 in “Continuous Groups— Lie’s theory with its applications to geometry, invariant theory, differential equations, systems of complex numbers, and non-Euclidean geometrics.” To this latter application of Lie’s theory Dr. Halsted devoted a considerable part of his “Report on Progress in Non-Euclidean Geometry” to the American Association for the Advancement of Science. Some idea of the growing and widespread interest in this modern development of science may be gained from the following extract from a circular written and circulated by Professor Wm. W. Payne, editor of “Popular Astronomy”: “Goodsell Observatory of Carleton College, “Northfield, Minn. “The Non-Euclidean Geometry “To the Teacher of Geometry: “Teachers of Elementary Geometry everywhere will be interested in the recent studies of the scholars in Pure Mathematics, at home and abroad, who have been investigating the claims of Non-Euclidean Geometry. “Large attention was given to this topic at the last meeting of the American Association for the Advancement of Science, at which Professor George Bruce Halsted, of the University of Texas, made a full report on this important theme. “Professor Halsted has consented to rewrite that scholarly paper in condensed form and plain language, especially for the benefit of Teachers of Geometry in High School, Academy and College, who want to know the latest views of eminent scholars of Mathematics in regard to the Non-Euclidean Geometry. “This knowledge will be of help to any one in teaching the elements of Geometry in any school.” Our loss in Dr. Dickson is Chicago’s gain. Two young men 34 CHAPTER 1. THE EARLY YEARS

of the same sort of promise, F. H. Smith and R. L. Moore, are this year showing that the splendid quality of Texas youth is of undiminished vigor, and as the School of Pure Mathematics has supplied the faculty of Yale and Chicago, so may it in the future be ready to give of its young vitality to Harvard and Princeton.28

Halsted’s last statement expresses obvious hope that one or more of is students would return to Halsted’s alma mater, Princeton. The road was opening for students to move from Texas to Chicago. Dickson had followed that path, going from Halsted to success at Chicago and then even greater success followed his graduation from Chicago. By August, 1899, Dickson already had much research in print. It is certain that Dickson’s move to the University of Chicago was a tremendous loss to the University of Texas, as indicated by the statement entered into the University Record in 1900.

The Faculty suffers the loss of two of its strongest members this year, both of whom voluntarily resign to accept positions offering wider fields of usefulness. Dr. Leonard E. Dickson was graduated from the University of Texas in 1893 with the degree of Bachelor of Science. During the following year he was fellow and graduate student in Mathematics. The thesis he submitted for the Master’s degree not only won that degree, but upon it he was awarded both a scholarship in Harvard and a fellowship in the University of Chicago. Going to Chicago, after two years of graduate study he received the degree of Doctor of Philosophy, magna cum laude. After spending a year in Europe, principally under the tutelage of the great German mathematician Sophus Lie, he was called to the University of California. He taught there for one year, and then accepted a call to serve his alma mater as Associate Professor. It was hoped that Dr. Dickson would be content to devote his life and talents to the University of Texas, but the opportunities for advancement, the wider field for work in the higher branches of mathematics offered by the University of Chicago, have proved too attractive, and he leaves to become Assistant Professor of Mathematics in that institution. His work in the University of Texas has been highly successful. Chiefly under his direction the bugaboo of freshman mathematics has lost many

28The University Record (The University of Texas, Austin, Texas, 1900) Vol. II, No. 2, June 1900, pp. 165-166. 35

of its horrors. He also taught a class during the year in the ﬁeld of his special interest—the Group Theory. With mathematical genius of high order, success is sure to attend him wherever he goes. In his upward progress his friends and associates here will always take an especial pleasure and pride.29

A mathematician, E. H. Moore, had been named Head of the Depart- ment of Mathematics at the University of Chicago and was developing a reputation of enticing to his department the most fertile of minds, both on the faculty and the student body. His hiring of Dickson provides such an example. Indeed, E. H. Moore is reported to have said of Dickson that: “Mr. Dickson was the most thoroughly prepared student in pure mathematics who had ever come to me.”30 The stage was set for Robert Lee Moore to come to the notice of E. H. Moore. Developments soon occurred which brought Robert Lee Moore forcefully to the attention of Chicago’s E. H. Moore. Even though no doctoral degree had been granted in mathematics at the University of Texas, it was Moore’s intention to remain there during the academic year 1902-03. He had just completed a year in which he was a Fellow in Pure Mathematics. He had been recommended for a position the next year as a Tutor in Mathematics. Halsted had recommended him for that position and both Halsted and Moore expected that it would be approved. Instead, the recommended appointment was not approved by the Board of Regents and Miss Mary E. Decherd was appointed to that position. Robert Lee Moore had much reason to expect approval of Halsted’s recommendation for his appointment as Tutor in Mathematics. By April 1902, Moore had made a substantial contribution as a mathematical researcher. He obtained a mathematical result which was to bring him to the attention of many, but particularly, E. H. Moore. However, he was to have already accepted a teaching post at a high school in Marshall, Texas before his opportunity at Chicago would present itself. That Robert Lee Moore did not receive a position, as Tutor in Mathe- matics, seemed to culminate growing discontent between Halsted and oﬃ- cials at the University of Texas. Indeed, the negative response to Halsted’s recommendation concerning Moore may have been directed more toward Halsted than toward Moore. Halsted had been quite an outspoken man at the university, and was a popular speaker. He had traveled much and was often invited to address others to describe his travels. Even within the classroom he often would treat subjects other than mathematics. This dealing with other than mathematical topics sometimes included comments

29ibid., p. 189. 30The CACTUS (University of Texas, Austin, Texas, 1896) p. 127. 36 CHAPTER 1. THE EARLY YEARS about other faculty members. One such statement, made in class before students, was about a member of the physics department: “He sits rocking on his front porch and thinks that he thinks.” Such comments as that surely would eventually make their way across to the subject of the comment and surely would be expected to give rise to acrimonious feeling toward Halsted. Halsted’s training had been from northern institutions: A. B., Prince- ton, 1875; A. M., Princeton, 1878; Ph.D., Johns Hopkins, 1879; then, teaching for three years at Princeton before coming to Texas in 1884. His youth surely was inﬂuenced by the Civil War and his outspoken manner, both within and without the classroom probably gave rise to some feeling that he was not altogether in sympathy with Southern attitudes or with some people who held to Southern attitudes. With his propensity to speak out about any topic of interest to him, and with some gift of brilliance, Halsted was the sort of person who would make many statements of a nature to cause feelings against him.* An Austin lawyer, upon being asked why Halsted left Texas, is reported to have said, “Well, Halsted just had more intelligence than the remainder of the faculty, taken together, and they just couldn’t stand it!” Because of a general feeling of discontent toward Halsted, from a variety of sources, by 1901, it perhaps was a means of discouraging his remaining at Texas by rejecting his recommendations. When his nomination of Robert Lee Moore as Mathematics Tutor was rejected in favor of Miss Mary E. Decherd, Halsted was incensed. He saw a gifted student, Moore, being discouraged by that action, so as to employ a woman who had established herself with those in authority, but who had little promise as a mathematician. Halsted, in attempting to have Moore continued at Texas, even raised the money for Moore’s salary, some ﬁve hundred dollars. The Presi- dent reacted the proposition. Thus it was, on October 24, 1902, after Miss Decherd had been formally appointed to her position on June 12, 1902, that Halsted had published his unhappiness with that decision. In SCIENCE MAGAZINE, in commenting on the Carnegie Institute, Halsted stated

. . . And if the keenest, brightest, most gifted of the young people reject the scientific career, then fellowships serve only a dull, stale, tired clique of incompetents. Even after the possession of the rare and precious gift of scientific genius has been clearly, competitively proven, the pos- sessor may choose what he considers a safer, more paying, more attractive career. I was twice Fellow of the Johns Hopkins Uni- versity and among my contemporaries, two unsurpassed in gifts for scientific creativity deliberately went over to moneymaking. 37

And finally among the sifted (sic) few who have the divine gift and the divine appreciation of their gift, the exquisite bud in its tender incipiency may be cruelly frosted. Of the great Hilbert’s ”betweenness” assumptions one was this year proved redundant by a young man under twenty working with me here, and by a demonstration so extraordinarily elegant and unexpected that letters from high authorities came congratulating the university on the achievement. Profes- sor E. H. Moore, of the University of Chicago, has published his congratulatory letter spontaneously written (Amer. Math. Monthly, June-July pp . 152, 153). This young man of marvelous genius, of richest promise, I recommended for continuance in the department he adorned. He was displaced in favor of a local schoolmarm. Then I raised the money necessary to pay him, only five hundred dollars, and offered it to the President here. He would not accept it. . . The bane of the state university is that its regents are the appointees of a politician. If he were even limited by the rule that half of them must be academic graduates, there would be some safety against the prostitution of a university, the broadest of human institutions, to politics and sectionalism, the meanest provincialism.31

Halsted ended his tenure as Professor of Pure Mathematics on December 10, 1902. On Thursday, December 11, 1902 there appeared in the Austin Daily Statesman the article following.

Dr. George B. Halsted, professor of mathematics at the State University, has severed his connection with that institution. This is not a surprise, in view of the fact that for several years past there have been differences between the professor and the regents and early last spring his withdrawal was forecasted in these columns. The differences arose several years ago by the fact that a general reduction of leading professorship salaries oc- casioned a reduction in that of Professor Halsted. At that time it was talked that he would resign. Later on another rumor was started that he would sever his connection with the institution, and last spring it became definitely known that a year’s notice had been given looking to the severance of relationship between the State University and Professor Halsted. The end has been

31Science Magazine, Friday, October 24, 1902, p. 645. 38 CHAPTER 1. THE EARLY YEARS

reached now in the withdrawal of Professor Halsted from the University. The announcement was made yesterday in a statement given by Regent Lomax to the press, reading as follows: “It is announced that Dr. George B. Halsted, professor of mathematics in the University of Texas will no longer be connected with that institution. At the recent meeting of the board of regents notice was given to him that his services were no longer required. It is also stated that Dr. Halsted received notice from the regents this past summer that he would be allowed to retain his position to the end of the present session, provided his services were satisfactory. His retirement at this time indicates that the board considered that it was for the best interest of the institution that his connection be severed at once. “No dissension exists in the present faculty, but on the other hand it is admitted on all sides that the relations between the authorities of the University and it’s faculty were never more harmonious”

So it was at the time Robert Lee Moore was about to make his way from the University of Texas to Marshall, Texas, with his mathematical career yet in its infancy, his ﬁrst mentor, Halsted, was experiencing diﬃculty at Texas, coming toward the end of his career there. The mathematical result which had so excited Halsted, and was to bring attention to R. L. Moore, as well as to Halsted, came about as a result of Halsted’s interest in geometry and correspondence he had with the renown German mathematician, Hilbert. Sometime in the early fall of 1901, Halsted was preparing his “Supple- mentary Report” to the A. A. A. S., (SCIENCE, No. 8, 1901) in which he treated Hilbert’s remarkable set of axioms for geometry.32 One group of assumptions made by Hilbert were called, by him, the “Axioms of Ar- rangements.” Halsted called them the “Betweenness Assumptions.” Of these assumptions, the fourth one stated:

Any four points, A, B, C, D, of a straight can always beso arranged that B lies between A and C and also between A and D, and furthermore C lies between A and D and also between B and D.

32Vorlesug ¨uber Eukledsche Geometrie (Festschrift, 1898). 39

Halsted, in translating from Hilbert’s statements, to arrive at the above, had come to an interpretation of “angeordnet.” That interpretation he discussed with R. L. Moore. Endeed, he asked Moore whether that fourth assumption might be demonstrated from the other assumptions. Those other assumptions, as translated by Halsted are:

If A, B, C are points of a straight, and B lies between A and C, then B also lies between C and A. If A and C are points of a straight, then there is always at least one point B, which lies between A and C, and at least one point, such that C lies between A and D. Of any three points of a straight there is always one and only one, which lies between the other two. DEFINITION. The system of two points A and B, which lie upon a straight α, we call a sect, and designate it with AB or BA. The points between A and B are said to be points of the sect AB or also situated within the sect AB; all remaining points of the straight are said to be situated without the sect AB. The points A, B are called endpoints sect AB. Let A, B, C be three points not co-straight and α a straight in the plane ABC striking none of the points A, B, C: if then the straight α goes through a point within the sect AB, it must always go either through a point of the sect BC or through a point of the sect AC.

Halsted had written to Hilbert, asking whether Hilbert “recognized any desirability for change.”33 Halsted was preparing to write a geometry text book, using Hilbert’s axioms as a basis for it. Hilbert’s answer to Halsted was received April 14, 1902 and read, in part, “Instead of II 4 (the fourth assumption), I believe it suffices simply to say: If B lies between A and C and C between A and D, then lies also B between A and D; and then to prove my old II 4 as theorem.”34 Halsted read this to Moore and suggested that Moore fill in the proof. Within the space of that single evening, Moore was able to inform Hal- sted that he had demonstrated Hilbert’s new axiom, eliminating II 4 and reducing “The Betweenness Assumptions” from five to four.* *Halsted stated that Moore had come to him early the next morning ”with his demonstration to Hilbert’s new axiom.” Burton Jones wrote (Proceedings of Emory Topology Conference 1970): “When

33American Mathematical Monthly, Vol. IX., (1902) p. 99. 34ibid. 40 CHAPTER 1. THE EARLY YEARS

Moore had discovered a redundancy in Hilbert’s axioms, and checked it through, it was after supper (and possibly on Sat- urday). Nevertheless, he had a strong urge to show it to Halsted immediately. Hurrying over to the campus he looked up toward the old main building. Years later, Moore remembered what a good feeling it gave him to see the light shining from Halsted’s window. Halsted was there.” Halsted was certain that Moore had no intimation that anyone had ever tried to prove those theorems. Moore started from the fact that some two weeks earlier, Hilbert thought an assumption necessary of which Moore oﬀered the demonstration on the basis of the remaining axioms. Halsted wrote Moore’s oral arguments and submitted them to the Math- ematical Monthly for publication. He also wrote E. H. Moore, who was chairman of the Department of Mathematics at the University of Chicago, telling E. H. Moore of R. L. Moore’s success reducing Hilbert’s axioms. No doubt, this was Halsted’s way of introducing E. H. Moore to another student of much promise. Dickson remained on the faculty at the University of Chicago and it would be diﬃcult for E. H. Moore to overlook the past good experience of accepting a student produced by Halsted. E. H. Moore replied to Halsted:

“I have received from you the April number of THE AMER- ICAN MATHEMATICAL MONTHLY, containing the proof by Mr. R. L. Moore of the redundancy of Hilbert’s Axiom II 4. The proof is certainly delightfully simple.”35

E. H. Moore also wrote to R. L. Moore:

The University of Chicago, May 6, 1902

Mr. R. L. Moore, The University of Texas, Austin, Texas. MR. DEAR MR. MOORE:ASK I read with much interest, the other day, your proof of the redundancy of Hilbert’s axiom II 4, in his system I, II, as exhibited by Professor Halsted in the current number of the AMERICAN MATHEMATICAL MONTHLY. Today I received from Professor Halsted a copy of that number. This is in response to a letter I sent him a week or so ago stating that I should be pleased to receive for publication in the Transactions the delightfully simple proof of the redundancy of which he wrote [had written] to me. I certainly

35American Mathematical Monthly, Vol. IX, No. 6-7, June-July, 1902, p. 148. 41

agree with him in this estimate of your proof. Apparently he has not called your attention to the fact that the redundancy was pointed out by me and proved in my paper, which I am sending under separate cover, on the proective36 axioms of geometry, published in the January number of the Transactions. In ac- cordance with correspondence with him, it was in connection with this paper of mine that he wrote to Hilbert and received Hilbert’s response which led to your work on the subject. You will see that it was my desire to survey the whole system of proective axioms, and to exhibit a new system, and, in that connection to show that Hilbert’s axioms I 4 and II 4 were in his system redundant, and moreover, to furnish a satisfactory account of the roles of the axioms I 3, 4, 5 which had been held by Schur to be redundant. As to the axiom II 4, you will see that, by considerations of the other linear axioms alone, and so in particular without the use of II 5, or of my axiom 4, I prove on page 151 that the axiom II 4 is a result of the statement 21 which statement is the statement of your theorem I. Thus to complete the proof of the redundancy of II 4, in Hilbert’s system, I should today make use of your proof of theorem I. The proof that I give, in that it involves my triangle transversal axiom 4, is necessarily much longer. I have supposed that you might be interested in understanding how your paper impresses me, and remain with considerable interest in the progress of your mathematical career, Yours very truly, (Signed) E. H. Moore

I suppose that the letter as a whole may be of value and interest to some readers of THE MONTHLY.37 Chicago, June 8, 1902.33 There is little doubt that R. L. Moore’s result caught E. H. Moore’s full attention. Only a few months earlier E. H. Moore had published a paper (“On the projective axioms of geometry,” Trans. Amer. Math. Soc., January 1902, Vol. III, pp. 142-168, 501) in which he also had established the redundancy of Hilbert’s “Axiom II 4.” The proof due to R. L. Moore was remarkably straightforward and clear. Hardly any other proof could have gained E. H. Moore’s attention more completely! 36What should this word be?? 37ibid., p. 152. 42 CHAPTER 1. THE EARLY YEARS

Thus it was that the Spring of 1902 came to a close with Robert Lee Moore having accomplished substantial mathematics, enough to ex- cite mathematicians of reputation elsewhere. His hopes of remaining at the University of Texas were dashed when Halsted’s recommendation to employ him as a Tutor were rejected and that position filled by Miss Decherd.* George Michael Decherd, an employee of the State Treasury Office in Austin, and Kate Thompson Decherd were the parents of four children who attended the University of Texas. Mary Elizabeth graduated with a BS in 1892 and began teaching in Austin High School. ”During the time she was teaching in the Austin High School, Miss Decherd’s interest in mathematics continued to grow. She kept in close touch with the De- partment of Mathematics in the University, taking from one to two courses each year. In 1897 she had completed the required amount of graduate work for her Master’s degree.” (University Record, Vol. IV, No. 4, Decem- ber 1902, p. 472) She was president of the Ashbel Society in 1892 and Vice President of the Alumni Association in 1900. In the summer of 1901 she did graduate work in the University of Chicago. Henry Benjamin Decherd received a BS from the university in 1896 and an MA in 1897. He received an MD from the Medical School in Galveston in 1900 and was on the faculty there for a number of years before leaving to go to the Wills Eye Hospital in Philadelphia. William Thompson Decherd received a B. Lit. from the University in 1899. In 1898 he received the ”T”. He was employed as a clerk in the Treasury Department in Austin. George Michael Decherd received the BS in 1901. He was editor of the Athenaeum Magazine in 1898. He received an MD from the Medical School in Galveston and returned to Austin to practice medicine in 1905. This led to Halsted’s raising funds with which to employ Moore. Upon the rejection of that proposal, Halsted angrily denounced the Board of Regents in Science Magazine, thereby leading to further deterioration of relations between Halsted and the Board of regents. Moore sought out a position and accepted one as a high school teacher in Marshall, Texas. E. H. Moore’s response to R. L. Moore’s work was heartening and strengthened R. L. Moore’s resolve to pursue his studies further in mathematics. However, arrangements which would take him to study at the University of Chicago were not worked out quickly enough for his enrollment there immediately after his departure from the University of Texas. An earlier graduate from Texas, under Halsted, L. E. Dickson, had already gone to the University of Chicago and had so impressed those there, particularly E. H. Moore, that another promising young mathematician, carrying Halsted’s strong recommendation, would be well received at Chicago. So it was that Robert Lee Moore left the University of Texas, having taught there his last year and as he left to go to Marshall, before continuing 43 in earnest his mathematical career, his mentor Halsted was nearing the end of his tenure at the University of Texas. Moore’s success as a mathematician was to quickly flame into brilliance, lighting his path upward as a comet, matched perhaps only by Halsted’s career trailing off into oblivion. 44 CHAPTER 1. THE EARLY YEARS Chapter 2

The Graduate and Post Graduate Years

Robert Lee Moore spent a year teaching in high school in Marshall, Texas. It was to be a year in which he had not much opportunity to grow math- ematically; he was removed from contact with faculty and fellow students who would pose problems of interest. Moreover, teaching in high school was its own kind of experience for a young man hardly older than some of the students he was to teach. It was not to be a totally wasted year, though. He would experience growth as a teacher, as well as increasing his own maturity. Moore was not yet 20 years of age when he began teaching at Marshall in the fall of 1902. Some of his students were almost as old. However, he had taught at the University of Texas and already had behind him the experience of teaching students older than himself. There had been left no doubt as to who held command in his classroom and in Marshall High School, as well, there was no doubt as to who ruled. The students in Moore’s classroom sat in long rows with girls on one side of the room and boys on the other. 1 He handled discipline by keeping the entire class after school if he felt some one or more of the students had behaved improperly.2 He was considered “high tempered” by his students and some delighted in tantalizing him. One day he came to school with his shoes tied with cord laces. After lunch he found on his desk a pair of shoe laces with verse attached:

1Telephone conversation with Mrs. Paul Hintz, Marshall, Texas, September 9, 1971. 2Telephone conversation with Mrs. Archie Wallace, Marshall, Texas, November 21, 1971.

45 46 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS

An eagle ﬂew from North to South// With Bobby Moore in his mouth;// But when he saw he had a fool,// He dropped him in the Marshall School.3 3

Moore never learned the identity of the author of that piece of verse. The student’s referred to him as “Bobby Lee,” but never when addressing him directly. One young lady, a member of a class which was “kept-in” after school for thirty minutes happened to have a watch. When thirty minutes had passed, she stood and announced it was time to go. Then she left, leading the class from the room, without asking Moore’s permission to leave. Moore didn’t like that at all, but he had already said that they should stay thirty minutes and that they had done. Examples of his particular style of classroom discipline are many. One of his students from Marshall High School recalls: “I had worked some problems on the blackboard and placed the chalk in the tray and turned to go back to my seat. For some reason, the chalk fell from the tray and hit the floor; perhaps I had flipped it with my dress or hand. When Mr. Moore heard it fall, he became very angry and told me to pick the chalk up and put it back in the tray. Now I had been taught at home to be obedient, but in this case, I didn’t think he was being fair. So I told him that I hadn’t dropped the chalk and I wasn’t going to pick it up. However, as a rule, all the students liked Mr. Moore. We found we could get on his good side by asking a lot of questions, so we did.”4 During the year 1902-1903, arrangements were made for Robert Lee to enroll the next year at the University of Chicago to pursue further his studies in mathematics. He left Marshall a not much stronger mathematician, with one more year of maturity, and still unmarried. The University of Chicago was a recently established institution, having begun in 1892. R. L. Moore had come to the University of Chicago in 1903 as a result of the interest and efforts of one of the outstanding educators of American history, a man by the name of Eliakim Hastings Moore. Be- fore R. L. Moore had left the University of Texas he had discovered a neat redundancy argument for a theorem in geometry and his mentor at the University of Texas, George Bruce Halsted, had that argument published in the Mathematical Monthly. Remarkably, E. H. Moore had already come to the same conclusion, and he had established redundancy of Hilbert’s Axiom II 4 and this was included in his paper, “On the proective axioms of geometry,” published in January 1902 in the Transactions of the Ameri- can Mathematical Society. Thus, it surely must have gained E. H. Moore’s

3Letter from Mrs. Inez Hughes, Marshall, Texas, July 29, 1971. 4Telephone conversation with Mrs. Paul Hintz, Marshall, Texas, September 9, 1971. 47 full attention when he learned of a young man from Texas who had independently established a theorem and, indeed, had oﬀered a neater, more elegant argument than had E. H. Moore. Moreover, there had already been an outstanding student who moved from the University of Texas to the Uni- versity of Chicago and, from there, on to great mathematical success. This had been Leonard Eugene Dickson, who had ﬁrst studied under Halsted at Texas. Dickson had been considered by E. H. Moore as the most thoroughly prepared student in pure mathematics who had ever come to him, so it was natural that another promising student from Halsted at Texas would cause quick excitement among the people at Chicago. R. L. Moore’s early success dealing with Hilbert’s geometry axioms served to immediately enhance his promise and to capture the imagination of the people at Chicago.

The University of Chicago had experienced an infirm and unstable beginning. It had been predated by another institution of the same name. The earlier institution began in 1857 as a Bap- tist institution though its government and sources of support were not exclusively denominational. Due to unwise financial decisions, indebtedness spiralled so that by 1886 the trust deed was foreclosed and instruction ceased. The editor of the denominational paper in Chicago wrote: “We can say for the Baptists of America that they will never again try to build up a great institution of learning upon borrowed money.” In 1885 an attempt was made to rescue the old university, but the Baptists, as a whole, were indifferent. In 1886 the Blue Is- land Land and Building Company which owned land in Morgan Park offered the backers of the university 20 acres of land and other assets. The conditions attached to the offer were that the backers secure $100,000 for endowment purposes, erect a building, and open the school by September 1888. One of the backers of the school was Thomas Wakefield Goodspeed who had been a student at the old university and it was through his efforts that John D. Rockefeller was persuaded to support the building of a new university. A prime candidate for the position as first president of the new university was William Rainey Harper. Harper had received his doctorate at Yale in 1875 at age 18 and at age 19 had become principal of the Masonic College in Macon, Tennessee. In his biography entitled William Rainey Harper, First President of the University of Chicago (University of Chicago Press, Chicago, 1928), Thomas W. Goodspeed wrote: “He found that he could 48 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS

teach what he knew, in a way that awakened and interested his students and gave delight to himself.” In 1876 Harper joined the faculty of Denison University and in 1879 went to Morgan Park as a member of the faculty of the Baptist seminary where he taught Hebrew. Then although he was urged to become associated with the new university in Chicago, he accepted a lucrative position at Yale. At the time he was considering the offer from Yale he was cautioned by Rockefeller “against binding himself to his professorship for more than three, or at most five, years. . . ” Harper kept in close touch with the proceedings in Chicago and indeed did become its first president.5

Once Harper had come to a decision to undertake the assignment at the University of Chicago, he proved to be extremely able at attracting other capable people to his institution. Indeed, some of those he brought to Chicago were themselves imbued with the characteristic of bringing other strong people to the university. Eliakim Hastings Moore was one of those people who was attracted to the University of Chicago. When the university opened in the autumn of 1892, E. H. Moore was 29 years of age and was, at that time the ﬁrst chairman of the Department of Mathematics. He held that post, beginning in 1892, as the acting chairman of that post.

E. H. Moore came to the university with an able background and already a substantial reputation. He had done his undergraduate study at Yale University, having graduated from there with his A. B. in 1883. Two of his friends, Horace Taft and Sherman Thatcher, had persuaded him to enter Yale. Taft was the brother of the President William Howard Taft and the other was the son of a professor at Yale. Before attending Yale, and while still in high school, E. H. Moore had come under the inﬂuence of Ormond Stone, then director of the Cincinnati Observatory. Moore had worked one summer for Stone. Stone’s interest in mathematics was substantial and, after he had become director of the Leader-McCormick Observatory of the University of Virginia, was later moved to Harvard and then to Princeton. While Stone’s interest in mathematics caused early corresponding interest on the part of E. H. Moore, it was Herbert Anson Newton, Professor of Mathematics

5HARPER’S UNIVERSITY: The Beginnings, Richard J. Storr (University of Chicago Press, Chicago and London, 1966) pp. 3-5, 18-19. passim. 49 at Yale, who inspired E. H. Moore in the spirit of mathematical research. E. H. Moore enjoyed an illustrious career as an undergraduate at Yale. He took three prizes in mathematics and one each in Latin, English, and astronomy. He was vale- dictorian of his senior class, and by that time had earned the nickname “Plus” from his classmates. After graduation in 1883 with his A. B., he earned his Ph.D. at Yale in 1885. Newton, being highly impressed by Moore, and wishing to encourage the promising young mathematician, financed a year’s study for him at Göttingenin Berlin in return for E. H. Moore’s promise to repay him at some future date. It was during his year in Germany that E. H. Moore learned that his ability was great enough to compete with mathematicians the world over. Moore met and worked with Widder, Schwartz, Klein, Weierstrass, and er. His association with them was friendly; it was indeed a rich experience for the young Amer- ican mathematician. Returning from Germany E. H. Moore served as an instructor in the Academy at Northwestern University, 1886-1887, and then was tutor at Yale University for the next two years. He returned to Northwestern in 1889 as Assistant Professor and was promoted to Associate Professor there in 1891. While at Northwestern he published four papers in the field of geometry and one concerning elliptic functions. His growing reputation brought him to the attention of Harper, who appointed E. H. Moore Professor and Acting Head of the Department of Mathe- matics in 1892. Four years later, recognizing the remarkable job E. H. Moore was accomplishing, Harper made him permanent head. E. H. Moore held that post until 1931. A major strength of E. H. Moore was that of attracting uni- versally brilliant scholars to the University of Chicago. From Germany he attracted Oskar Bolza and Henrich Maschke, who received their Ph.D.’s from Göttingen. All three men, Moore, Bolza, and Maschke, were enthusiastic and totally dedicated to mathematics. Moore was brilliant in his scholarship and aggressive in his leadership. Bolza was “rapid and thorough and Maschke was more deliberate, but sagacious, and a delightful lecturer on geometry.” These three had no precedent to follow at the University of Chicago and, moreover, not much precedent anywhere in the country to guide them in the formation of their doctoral program. By attracting graduate students of 50 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS

high capability and by exciting these to effort of high order in their own mathematical research efforts, they soon established the University of Chicago as a center of mathematical activity which had much influence on the creativity of the American Mathematical school. E. H. Moore was influential, not alone by way of his mathematical research, nor even by way of his skill at gathering to Chicago people of much talent, but perhaps his greatest effect was by way of his teaching effort and style. He defied and violated most of the classroom procedures of pedagogy which were then accepted. He was totally absorbed in mathematics and was consumed with his efforts toward precision, clarity, exactness, and logical correctness.6

E. H. Moore’s influence on others about him, in his devotion to those characteristics, was apparent in many ways. Once, when R. L. Moore had been late for an appointment with E. H. Moore, he began to apologize by saying that the alarm clock did not go off at the prescribed time. Then, catching himself, being aware of E. H. Moore’s concern about totally accurate statements, he hurried on to say that, it is possible that the alarm did go off but that he did not hear it. But then he felt obliged to add that it may have gone off and he may have heard it but, upon turning it off, had fallen back asleep and then upon awakening earlier, not recalled it having ever gone off. At about this point, E. H. Moore relieved him of further explanation and they proceeded in their discussion.7 R. L. Moore already had some appreciation for such correctness of statements before coming to study with E. H. Moore. At one point, while taking an examination in his undergraduate days, he was confronted with an “honor system” statement, offering assurances that he had neither given nor received assistance during the period of the examination. Upon completing the examination, he sat down to write his statement, not finding it proper to simply sign the statement tendered him. He began by stating that at one point during the examination he rose from his seat, walked across the room to a window and raised it. He clearly did this for his own benefit and it may have been that such an act did offer assistance to some other person being examined. Indeed, his passage across the room may have distracted some from their work, or even others may have been distracted enough to lose a train of thought that was not fruitful so that upon their reapplication of their attention to their work, they happened upon a more fruitful direction

6Biographical Memoir of Eliakim Hastings Moore, G. A. Bliss and L. E. Dickson (National Academy of Sciences, Washington, D. C., 1936) passim. 7Personal conversation with R. L. Moore, Austin, Texas. 51 of thought. Knowing R. L. Moore’s concern for clarity and correctness of statements, the instructor interrupted him to declare further explanation unnecessary.8 There was much of a common feeling between E. H. Moore and R. L. Moore. E. H. Moore was thought by some to have had a slight speech impediment, or else to have been a person who had overcome a speech impediment of some earlier time. This arose from his speaking habit which was designed to allow accurate description of what he wished to state. It would seem at times that he would begin to say the word, testing it almost to see if it would convey what he desired, and determining that it would not, switch, in mid-breath, to another word. His classes were of indeterminate length, as he would bring before his students some topic which he found of absorbing interest and he, and they, would investigate those ideas until the topic for that day was exhausted. Formal class schedule, time of day, nor meal-time had any effect on his pursuit of the topic of the day. At times, because of his total absorption in the subject at hand, he would be unaware offense had been given by something he had said, and would be extremely gentle in his expressions of regret when it was called to his attention that someone’s feelings had been hurt by his impatience. Weak students often shunned his classes because of the demand placed on them. Strong students and those of quickest mind were attracted to him. He became a teacher who taught the teachers of mathematics. His teaching skills were the strongest when dealing with graduate students, although with students not so far advanced he was less accessible. His own mind was so quick and his concentration so thorough that it was difficult for him to await the comprehension and the slower development of understanding among those who were less experienced. However, he did edit an arithmetic book for use in elementary schools in 1897, and in 1903-04 and following years, he influenced radically the methods of undergraduate instruction in mathematics at the University of Chicago. He gave courses in beginning calculus himself, casting aside textbooks, and concentrating instead on the fundamentals of the topic and their graphical interpretation. During this period of time his classes met for two hours each day and required no outside work from the students. Obviously, much work was required of the instructor. The system was later abandoned because the two

8Personal conversation with R. L. Moore, Austin, Texas. 52 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS

hour demand on the student’s day each day oﬀered too many conﬂicts with the scheduling of other departments.9

In addition to Bolza and Maschke, E. H. Moore also had arranged for Oswald Veblen to stay on at the University of Chicago, following the completion of his doctoral work there. Veblen was only two years older than R. L. Moore, having been born on June 24, 1880.

He (Veblen) earned his A. B. degree in 1898 from the Univer- sity of Iowa, taught there as a laboratory assistant in physics for one year, and then went to Harvard where he received a second A. B. degree in 1900. From there he proceeded to the Univer- sity of Chicago where he studied with E. H. Moore for three years. At that time there was much interest in the foundations of Euclidean geometry. Euclid’s elements had been accepted as a model of logical precision. It had been shown logically inadequate because of its neglect of the order relations between points on a line and its consequent incapability to allow a rigorous proof that the plane is separated into two halves by a line into an inside and an outside by a triangle. Hilbert had proposed other supposedly more precise axioms which were in vogue. Those axioms depended upon primitive concepts, including point, line, plane, congruence, and betweenness. Veblen, in his thesis, took up another line of thought which had been started by Pasch and Peano that geometry was based directly on notions of point and order. In Veblen’s system then there were only two primitive notions, point and order (the points a, b, c occur in the order abc). When Veblen completed his thesis in 1903 he became an associate in mathematics at the University of Chicago and immediately became active in the training of other young mathematicians. Veblen recognized the importance and fundamental position of the Heine-Borel theorem in analysis. The idea underlying the Heine-Borel theorem was to express the compactness of the interval on a line and, more generally, is the fundamental concept underlying the study of compact topological spaces. 10

That same theorem was to become of central importance in a junior level course R. L. Moore developed at the University of Texas, which he

9Biographical Memoir of Eliakim Hastings Moore, G. A. Bliss and L. E. Dickson (National Academy of Sciences, Washington, D. C., 1936) passim. 10Biographical Memoir of Oswald Veblen, Saunders MacLane (National Academy of Sciences, Washington, D. C., 1964) pp. 324-341, passim. 53 called Introduction to the Foundation of Analysis. Many years, in that course he would raise the following four questions:

1. If G is a collection of intervals covering the interval [A, B], then some ﬁnite subcollection of G does.

2. If G is a collection of segments covering the interval [A, B], then some ﬁnite subcollection of G does.

3. If G is a collection of intervals covering the segment (A, B), then some ﬁnite subcollection of G does.

4. If G is a collection of segments covering the segment (A, B), then some ﬁnite subcollection of G does.

Students, in their attempts to decide whether any of these statements were true, would be compelled to discover for themselves the same fundamental property that Veblen back in 1903 and 1904 realized occupied a central position in analysis.

As head of the Department of Mathematics, E. H. Moore exercised very little control over the techniques of his colleagues. They were unlimited in the freedom to develop their own techniques. He only demanded success and always was sympathetic to new proposals and methods of instruction short of prescribing a textbook. It was to be expected that a man so highly regarded as a sci- entist should become a leader in his university and in the as- sociations of workers in his field. Professor Moore was one of the youngest, but also one of the most spirited, of the notable group of scholars who in the nineties of the last century first shaped the character of the new University of Chicago and gave it great distinction. From the opening day of the University he devoted himself unselfishly to its interests, and his counsel through the years had great influence. At all times he stood unequivocally for the highest ideals of scholarship. His services to the University were signalized in 1929 by the establishment of the Eliakim Hastings Moore Distinguished Service Professor- ship, one among the few of these professorships which have been named in honor of members of the faculty of the University. The first and present incumbent is Professor Leonard Eugene Dick- son. Professor Moore was a moving spirit in the organization of the scientific congress at the World’s Columbian Exposition of 54 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS

1893, and in the first colloquium of American mathematicians held shortly thereafter in Evanston with Klein as the principal speaker. He was influential in the transformation of the local New York Mathematical Society into the American Math- ematical Society in 1894, and in the foundation of the first so- called section of the Society whose meetings were held in or near Chicago and of which he was the first presiding officer in 1897. The formation of the Chicago Section was an outgrowth of the Evanston colloquium. After that meeting a number of mathematicians from universities in and near Chicago occasionally met informally for the exchange of mathematical ideas. After the organization of the American Mathematical Society they applied for and were granted recognition as a section of the So- ciety. It was the success of this first section which led to the establishment, in various parts of the country, of other similar meeting places which have added greatly to the influence and value of the Society. Professor Moore was vice-president of the Society from 1898 to 1900, and president from 1900 to 1902. In 1921 he was president of the American Association for the Ad- vancement of Science. In 1899 he and other aggressive members induced the Society to found the Transactions of the American Mathematical Society, now our leading mathematical journal. The first editors were E. H. Moore, E. W. Brown of Yale, and T. S. Fiske of Columbia. These men set standards of editorial supervision which have endured to this day. Professor Moore retired from his editorship in 1907. From 1908 to 1932 he was a non-resident member of the council of the Circolo Matematical di Palermo and of the editorial board of its Rendiconti. From 1914 to 1929 he was the chairman of the editorial board of the University of Chicago Science Series. Nineteen volumes were published in the Series during that period, two of them, by H. F. Blichfeldt and L. E. Dickson, in the domain of mathematics. ¿From 1915 to 1920 Professor Moore was a member of the editorial board of the Proceedings of the National Academy of Sciences.

In 1916, by his advice and encouragement, he gave great assistance to Professor H. E. Slaught, who was a moving spirit in the formation of the Mathematical Association of America. In the decades preceding 1890 research scholars in mathematics in America were few and scattered, with limited opportunities for scientiﬁc intercourse. At the present time we have a well- 55

populated and aggressive American mathematics school, with frequent opportunities for meetings, one of the world’s great en- ters for the encouragement of scientific genius. From the record of Professor Moore’s activities described above, it is clear that at every important stage in the development of this school he was one of the progressive and influential leaders. That the distinction of Professor Moore’s services to science and education was recognized in other universities as well as his own is indicated by the honors conferred upon him. He received an honorary Ph.D. from the University of Göttingenin 1899, and an LL.D. from Wisconsin in 1904. Since that time he has been awarded honorary doctorates of science or mathematics by Yale, Clark, Toronto, Kansas, and Northwestern. Besides his memberships in American, English, German, and Italian mathematical societies, he was a member of the Ameri- can Academy of Arts and Sciences, the American Philosophical Society, and the National Academy of Sciences. Two funds have been established in his honor. The first is held by the Amer- ican Mathematical Society for the purpose of assisting in the publication of his research and for the establishment of a permanent memorial to him in the activities of the Society. The second has been expended for a portrait of him which hangs in Bernard Albert Eckhart Hall for the mathematical sciences at the University of Chicago. The interest in these funds among the friends and admirers of Professor Moore was a remarkable tribute to him scientifically and personally.11

R. L. Moore, while at the University of Chicago, had exposure to several of the most able mathematical minds in the country. As well, he was exposed to various teaching styles, with the overriding demand imposed by E. H. Moore; that of doing whatever would bring best results, without regard to established methods. Maschke was thought by many to be a sound and delightful lecturer. At times it would seem that he would “smack his lips” over his own presentation, stepping back from the board and concluding his sentences with an audible smacking sound. It would happen that, as R. L. Moore would sit in Maschke’s class, thinking about problems of more interest to himself, Maschke would step back, smack his lips seemingly in appreciation over what he had just uttered, notice R. L. Moore’s lack of attention and ask, “Mr. Moore, is that clear?” R. L. Moore, with his

11Biographical Memoir of Eliakim Hastings Moore, G. A. Bliss and L. E. Dickson (National Academy of Sciences, Washington, D. C., 1936) passim. 56 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS devotion to correct statements would reply, “No, sir, to tell the truth, it isn’t,” for he had not been observing the proceedings enough to follow the argument. “Well, then, Mr. Moore, can you put your finger on the point that is giving you difficulty?” Maschke would continue. Then R. L. Moore would put an end to the conversation by saying something to the effect, “That’s all right, Professor Maschke, I’d rather that you wouldn’t bother. I’ll work it out for myself.” Imagine the reaction of a gifted lecturer, proud of his elocution and clarity, who is told by a student something to the effect that the student would really rather work it out for himself! While R. L. Moore was at the University of Chicago, E H. Moore was teaching by concentrating on the fundamentals of the topics and their graphical interpretations. Often he would cast aside the textbooks to proceed along a path he felt more productive. E. H. Moore developed a treatment of properties of the real line, proceeding from an axiomatic base. In particular, he took as fundamental axioms:

1. For every inﬁnite sequence of intervals aν . . . bν (ν = 1, 2, 3 ...) having the following properties: (a) every interval contains the succeeding intervals,

(b) the lengths of the intervals taken positively or an bn , decreases indefinitely as the number of intervals increases| 12−, then| there is one and only one point lying on each interval. 2. Dedekind’s Axiom: Suppose the complete line be divided into two sets of points, “left” and “right” such that every point (equivalently, number) of the left set is at left of (equivalently, less than) every number of the right set. This separation of the points is called a “cut.” Then in every cut of the linear scale there is either a right most left point or a left most right point and not both.13 E. H. Moore pointed out that the two axioms were equivalent and also explained the meaning of their being equivalent, stating that two statements are equivalent “if when one is assumed the other can be proved.” E. H. Moore defined limit point by stating: If every interval, I on the line, no matter how small, contains also a point xI S such that xI = x, ∈ 6 12This means that for every positive number there are intervals less than 13Microfilm copy of classroom notes by E. H. Moore, University of Chicago Library, Chicago, Illinois. 57

then x is a limit point of S.He called x an accumulation point of S if S is dense at x.

He would establish that a point set S, on a finite interval a...b, containing an infinitude of points has a limit point on the interval a...b and moreover, it has a definite left most and a definite right most limit point. The manner in which E. H. Moore approached his teaching of mathematical concepts had distinct influence on R. L. Moore. E. H. Moore offered two definitions of limit of a function, one analytic and the other geometric. For his analytic definition he stated, “the limit on the x set S for x = a of the function f(x) is a certain constant f means that: there exists a constant f such that for every positive number there exists a positive number δ depending on such that f(x) f < for every x such that x S, x = a | − | ∈ 6 and x a < δ. His notation for this was: | − | x S x, x | x = a x a < δ 6 | − |

His geometric definition was almost precisely the one which R. L. Moore employed in his definition of continuity of a simple graph (a simple graph is a point set on the plane such that no vertical line intersects it twice): The statement that the simple graph G is continuous at the point A of G means that if α and β are horizontal lines with A between them then there exist vertical lines h and k with A between them such that every point of G between h and k is between α and β. In fact, much of the content which R. L. Moore offered in his course “Introduction to the Foundations of Analysis” at the University of Texas was seen by R. L. Moore perhaps first win E. H. Moore or Oswald Veblen. Indeed, in the course which R. L. Moore later developed as one in which he uncovered much talent, he took as his basic axiom a version of Dedekind’s: If each of S1 and S2 is a set such that S1 +S2 is the real line and each point of S1 is to the left of each point of S2 then either S1 has a right most point or S2 has a left most point. One of the major theorems in that course was stated: If M is an infinite subset of a segment then M has a limit point. Still another was reminiscent of E. H. Moore’s axiom which was equivalent to Dedekind’s axiom. R. L. Moore stated the theorem thusly: If each of [al, bl], [a2, b2], [a3, b3],... is an 14 interval such that for each positive integer n,[an+l, bn+l] is a subset of [an, bn] then there is a point common to all the intervals and that common part is either an interval or a point.

14“Interval” in R. L. Moore’s usage is closed. “Segment” does not include the end points; the interval does contain its end points. 58 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS

While it is clear that certain mathematical concepts are difficult to overlook as one prepares the content of a certain course, it still remains true that R. L. Moore, in deciding which concepts he would introduce in his Introduction to the Foundations of Analysis, hearkened back to his own early experience in which he was introduced to the foundations of analysis. The choice of the statement of his fundamental axiom would be reminiscent of that used by E. H. Moore and the choice of the statements of his theorems a mixture of E. H. Moore and Oswald Veblen. It was Veblen who was credited with an early appreciation of the central role of the “covering” theorem, as R. L. Moore was to call it, in analysis. Thus it would be that the subject matter which was of considerable attention in the early 1900’s at the University of Chicago would be the same subject matter which R. L. Moore would modify and restate so that he would gain increasingly better opportunity to discover and develop the talents and abilities of his students. Indeed, it was in Veblen’s dissertation, in which he gave a system of axioms for geometry, that the so-called Heine-Borel property appeared as Axiom XI: If there exists an infinitude of points, there exists a certain pair of points AC such that if σ (σ denotes a set or class of elements, any one of which is denoted by alone or with an index or subscript) is any set of segments of the line AC, having the property that each point which is A, C or a point of the segment AC is a point of a segment σ, then there is a finite subset σ1, σ2, . . . , σn with the same property. Veblen called Axiom XI his Continuity Axiom and noted that Schoen- flies15 had called that property the Heine-Borel theorem. Borel is given credit for first stating the theorem16 in 1895, as a theorem of analysis, but Heine in 1871 used that notion in the proof of a theorem of uniform continuity.17 Veblen credited the idea of the equivalence of his Axiom XI with the Dedekind cut axiom as resulting from a conversation between himself and N. J. Lennes.18 It is apparent from Veblen’s dissertation that there was much ! exciting discussion occurring at the University of Chicago during those 3 years of the early l900’s. Although Veblen was a more mature graduate student than R. L. Moore, the excitement over the developments in geometry was a commonness which caused them to function, each to the other, as a catalyst. Indeed, footnoted in Veblen’s dissertation is the statement, “I

15A system of axioms for geometry, Oswald Veblen, Trans. Amer. Math. Soc., 5 (1904) 347-348. 16E. Borel, Annuales de l’Ecole Normal Superieure (3) Vol. 12 (1895) p. 51. 17E. Heine, Die Elemente der Functionlahre, J. Reine und Ungewante Math. (Crelle) , Vol. 74 (1872) p. 188. 18A system of axioms for geometry, Oswald Veblen, Trans. Amer. Math. Soc. 5 (1904) 348. 59

wish to express deep gratitude to Professor E. H. Moore who has advised me constantly and valuably in the preparation of this paper, and also to Messrs. N. J. Lennes and R. L. Moore, who have critically read parts of the manuscript.” Veblen stated as axioms:

Axiom II: If points A,B,C are in the order ABC, they are in the order CBA. Axiom III: If points A,B,C are in the order ABC, they are not in the order BCA. Axiom IV: If points A,B,C are in the order ABC, then A is distinct from C

and then refers to R. L. Moore again, stating, “Mr. R. L. Moore suggests that ’If A is a point, B is a point, C is a point’ would be a more rigorous terminology for the hypotheses of II, III, IV, inasmuch as we do not wish to imply that any two of the points are distinct.” It had happened that E. H. Moore and R. L. Moore each proved a redundancy concerning Hilbert’s Axiom II 4. Surely, R. L. Moore found himself quickly in the center of a faculty excited over new developments in geometry. Veblen was stimulated by E. H. Moore to develop a system of axioms for geometry, taking as undefined “points” and “order,” thus following the trend of Pasch19 and Peano20 instead of Hilbert 21 or Pieri. 22 Veblen wrote his dissertation, “A system of axioms for geometry,” under the direction of E. H. Moore. The topics treated in his dissertation were certainly of interest and were topics of discussion among others at Chicago. Deeper in Veblen’s dissertation is a theorem which merited the author’s comment: “This proposition was given as ‘Axiom II 4’ in Hilbert’s Festschrift (p. 7). Its redundancy as an axiom of Hilbert’s system I, II was proved by E. H. Moore, On the projective axioms of geometry, Transactions of the American Mathematical Society, January, 1902, vol. 3, p. 142-168, 501. A second proof has been given by Mr. R. L. Moore, cf. p. 98, American Mathematical Monthly, April 1902.” Veblen’s dissertation dealt with a system of axioms which are stated in terms of a class of elements called “points” and a relation among points called “order.” He followed the trend of development inaugurated by Pasch and continued by Peano rather than that of Hilbert or Pieri. Veblen, in the statement of his Axiom XI, recognized the importance of the Heine-Borel Theorem, stating: 19M. Pasch. Vorlesungen über neuere Geometrie, Leipzig, Teubner, 1882. 20G. Peano, I principii di geometria, Turin, 1889, Sui Fondamenti della geometria, Rivista di Matematica, Vol. 4 (1894) pp. 51-59. 21D. Hilbert, Grundlagen der Geometrie, Leipzig, 1899. 22M. Pieri, Della geometria elementare come sistema ipotetico deduttivo. Mongrafia dei punto e dei moto. Memorie della Reale Academia delle Scienze di Torino (2), Vol. 49 (1899) pp. 173-222. 60 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS

The proposition here adopted as the continuity axiom is referred to by Schoenflies as the Heine-Borel theorem. So far as I know, it was first stated formally (as a theorem of analysis rather than of geometry) by Borel in 1895 but is involved in the proof of the theorem of uniform continuity given by Heine in 1871. The idea of its equivalence with the Dedekind cut axiom was the result of a conversation with Mr. N. J. Lennes.23 Thus it was that R. L. Moore was intimately informed as to Veblen’s dissertation. It dealt with subject matter which Moore had already been concerned with while still at Texas and he clearly was involved in discussions with Veblen during and before the writing of the dissertation. A natural development, particularly in the light of E. H. Moore’s encouragement of Veblen to early involve himself in direction of students, was Veblen’s suggesting topics to R. L. Moore for investigation. Resulting was R. L. Moore’s dissertation, Sets of metrical hypotheses for geometry. Indi- cation of R. L. Moore’s concern for correctness of statements is given by a paragraph appearing early in his dissertation: In this paper “proof” is used to mean an indication of a demonstration. “Hence,” “by,” “therefore,” etc., are intended as suggesting certain relations and not necessarily as describing exactly the logical dependence of one statement upon another.24 In his dissertation, R. L. Moore gave a set of assumptions concerning point, order, and congruence which together with other order assumptions, a continuity assumption, and a weak parallel assumption, were sufficient for the establishment of ordinary Euclidean geometry. While Veblen’s work seemed to be drawing him toward a notion of non- metric considerations, R. L. Moore’s work seemed to be more directed toward the metric situation. In fact, Veblen shortly published (1905) a paper entitled “Theory on plane curves in non-metrical analysis situs,” in which he argued a proof of the fundamental theorem proposed by Jordan: A simple closed curve lying wholly in a plane decomposes the plane into an inside and an outside region. The setting for this theorem was taken by Veblen to be a space satisfying axioms I-VIII, XI of his dissertation, stating explicitly that nothing is assumed “about analytic geometry, the parallel axiom, congruence relations, nor the existence of points outside a plane.”25 23A system of axioms for geometry, Oswald Veblen. Trans. Amer. Math. Soc. 5 (1904) 348. 24Sets of metrical hypotheses for geometry. Trans. Amer. Math. Soc. 9 (1908) 487-512. 25Theory of plane curves in non-metrical analysis situs. Trans. Amer. Math. Soc. 6 (1905) 83-98. 61

Following R. L. Moore’s graduation from the University of Chicago, he continued to ponder Veblen’s axioms and the conclusions drawn from them. By 1907, he published “A note concerning Veblen’s axioms for geometry,” in which he offered some improvement of them. However, it was several years later, in 1912, that R. L. Moore obtained rather devastating results concerning Veblen’s non-metrical analysis situs. In “On a set of postulates which suffice to define a number-plane,” R. L. Moore established that any plane satisfying Veblen’s Axioms I-VIII, XI, of his “Systems of axioms for geometry,” is a numberplane. That is, any plane satisfying those axioms contain a system of continuous curves such that, considering those curves as straight lines, the plane is an ordinary Euclidean plane. Thus, Moore claimed that any discussion of analysis situs based on Veblen’s axioms (as for example Veblen’s proof of the theorem that a Jordan curve divides its plane into just two parts) is no more general than one based on analytic hypothesis. The results obtained by Moore were remarkable and unexpected by many. The mathematics involved was difficult in that Moore established a one-to-one reciprocal continuous correspondence between the continuous curves of Veblen’s planes and straight lines of an ordinary Euclidean plane. If Robert Lee Moore had not a substantial reputation before publicizing those results, he surely had established himself after those results were known. 62 CHAPTER 2. THE GRADUATE AND POST GRADUATE YEARS Chapter 3

The Formative Years

The years following graduation from the University of Chicago were formative ones for R. L. Moore. He was able to devote himself to his teaching and his mathematics. His first appointment was as Assistant Professor at the University of Tennessee. His stay there proved to be rather non-productive. A partial reason may have been his health. The summer following completion of his work at Chicago and prior to going to Tennessee, he was visiting at home in Dallas. He and a brother went camping sometime during the summer and he contracted malaria. He jaundiced and his health was not sound as he went to Tennessee. The department at Tennessee surely must have been considerably below his experience at Chicago and, indeed, Texas. There were only two faculty, Schmitt the chairman and Moore. The undergraduate offerings were no more sophisticated than calculus. The graduate offerings were described as follows in the 1906-1907 catalog:

a. Higher Algebra. In this course a knowledge of all the preceding courses is presupposed and the principles of algebra are viewed in their widest applications. Chrystal’s Algebra and lectures. Senior and Graduate. Fall. Two hours. b. Trigonometry of Imaginaires. A course embracing Demoivre’s Theorem, hyperbolic trigonometry, and many other subjects not included in courses four and ﬁve. Lock’s Higher Trigonometry. Senior and Graduates. Winter. Two hours. c. Modern Geometry. The principles of continuity, duality, inversion, harmonic ranges, poles and polars, and other subjects are considered. Taylor’s Euclid, V-VI. Senior and Graduate. Spring. Two hours.

63 64 CHAPTER 3. THE FORMATIVE YEARS

Should there be demand therefore, one or two of the following courses will be given in 1906-1907: d. Foundations of Arithmetic and Geometry. Including a consideration of various sets of postulates and a study of relations existing between fundamental propositions. (In particular a discussion of the genesis of the ordinary number system from postulates concerning positive integers.) Pre- requisite: A certain aptitude for close critical reasoning. e. Non-Euclidean Geometrics including the Bolyai- Lobachevshian and Riemannian geometries. f. Theory of Functions of a Real Variable, including a study of limits and continuity. g. Theory of Functions of a Complex Variable. It is desirable that there should have been a previous careful study of limits and continuity.

No doubt Moore experimented with his teaching approach and this may not have been well received in all quarters. Moreover, there did occur a difference between Moore and the chairman. Following the end of the first semester Moore was slow to turn his grades in to the appropriate office. In fact, he had not even begun to grade the papers when he was approached on the campus one day by the chairman, Schmitt, who inquired, “Professor Moore, I notice that you have not yet turned in your grades. Have you finished grading the papers?” Moore replied that he had not, but did not elaborate further; that is, that he had not even begun. Sometime later the chairman became aware that, at their earlier meeting, Moore had not even begun grading the papers. He called Moore in and accused Moore of lying to him.1 This, of course, was not received with the happiest of feelings by R. L. Moore and may have been a major factor in his decision to leave Tennessee. However, his decision to leave Tennessee may have come simply because he felt that there was not faculty or student potential enough to merit his sustained effort. It is not surprising that the offerings described in the 1906-1907 catalog, to be offered if interest developed, were not offered. Moore had left and the beginning of the 1906-1907 academic year found him at Princeton. One other alteration had occurred during the year at Tennessee. Moore had grown a mustache, and though he shaved it before returning home, he did send home a picture. He felt his father was pleased by it; indeed, his father had been full-bearded as long as Robert Lee Moore could remember.

1Personal conversation with W. L. Ayres, Dallas, Texas, April 11, 1972. 65

Halsted had been away from Texas several years by 1906, but he no doubt learned of Moore’s move to his own alma mater. There is some reason to think that Halsted yearned to see a successful protege of his take a position at Princeton. R. L. Moore had been preceded to Princeton by Oswald Veblen, who went there as Preceptor the same year Moore had gone to Tennessee. The preceptor system is attributed by some to Woodrow Wilson. Our class was the one who had fortune to have the Preceptors, fifty of whom joined the teaching faculty. In some courses they met a small group of students, not more than ten, and we met in rooms on the campus or in their living quarters. We would discuss with our preceptors about our assignments. I think my class did much better with that situation than to be in larger rooms with a large number of students. Many of these preceptors were promoted and many of them were well know while many of them remained at Princeton. Some of the mathematics teachers - some of them arranged to have some of them attend a room in the evenings. I frequently consulted with them, so that I learned enough to pass any examination I took.2 My four years at Princeton University were under Woodrow Wil- son. My class was the first to receive and have four years of his Preceptoral system with which you are, no doubt, familiar. It has something of the Socratic system in it. The President brought 50 new men to Faculty recruited from all over the U. S. As Preceptors they also had classes. The Prof in each course had us in the classroom, but one hour a week was subtracted from hours of courses and given to a Preceptor who had no more than five men in his Study or room for that purpose in recitation building. In this informal atmosphere we discussed the course, were assigned some outside reading on which we were asked questions later. So we had two men for every course and it was helpful and stimulating. Many of the original 50 rose from the rank of Assistant Prof. to Heads of Departments and some to becoming Deans. Princeton still clings to this system. It required a larger Faculty and that is why Princeton with a smaller number of students than Yale and Harvard has a larger Faculty than either of them.3 Wilson had attracted some fifty men to Princeton at the outset of his 2Letter received from Mr. William N. Ottinger, Philadelphia, Pennsylvania, October 1971. 3Letter received from Mr. Wallace H. Carver, Caldwell, New Jersey, May 1971. 66 CHAPTER 3. THE FORMATIVE YEARS presidency at Princeton. They served as preceptors; that is, as individuals who taught classes and who also served as assistants in other classes. In those instances, a professor would have his usual weekly class time reduced by one hour. During that hour students of his class would meet with the preceptor in the preceptor’s study or room to discuss the ideas of the course, or else to discuss readings which had been assigned earlier in the regular class. No preceptor was assigned more than five or ten students. Princeton thus early established a low student-faculty ratio and, to this day, has a sizeable faculty, for the number of students there, if compared to many other universities. Thus it was that Oswald Veblen, and then a year later, Robert Lee Moore, went to Princeton as Preceptors. However, the Princeton system was not as individualized as might be suggested by the concept of the preceptor. Indeed, many of the courses were taught in more than one section, but were required to cover the same material. This requirement came by way of “departmental” examinations which were administered to each section, regardless of instructor. Moreover, the several instructors would collect together to grade the examinations en masse, with each grader having responsibility for specific problems. This surely reduced the effect of the day-to-day evaluation by the teacher. Moreover, the system probably caused some to teach “toward” the examination, instead of striving to develop the student’s skills or abilities. Although R. L. Moore had held his doctoral degree for only one year, prior to coming to Princeton, this was not a philosophy consistent with his experience as a student of mathematics. Neither did it fit with his desires to teach that subject in such a fashion. The attitudes of some in the mathematics faculty grated on him. In one instance, a large class was having a test administered and each student had been instructed beforehand to bring a “blue-book” instead of loose- leaf paper. After the test had begun, one of the faculty proctors stopped in front of the desk of a student, reached down to remove a blue-book and announced in a very loud voice, “Only one blue-book!” It was the sort of person who would do that sort of unnecessary thing that disturbed Moore. It is not surprising that Moore sought association outside the faculty of the mathematics department. One of his most highly regarded associates was “Spider” Relley, the boxing instructor at Princeton. Moore was an enthusiast for competitive activity, though to interest him, it needed to be individual in nature. Boxing fit Moore’s requirement exactly, and he took to that sport with remarkable dedication. Then, too, as he came to Princeton, he was recovering from malaria and physical exercise would tend to overcome the residue of his illness. He was never a large man, though he was quite active. Shortly after going to Princeton, his weight was recorded at 143, though he weighed as much as 165 before he left there. The boxing 67 he did at Princeton was of a sparring nature, not professional, and with whomever might be at the gym and wanting to spar. The procedure was that two boxers would agree to spar and they would decide whether to “go hard” or “go easy.” While they would not attempt to knock each other out, a decision to “go hard” simply meant that they would not pull their punches. Since they had no one to time their rounds, they would spar until one of them called “stop!” Moore never felt that he should call “stop” but rather, that he should let the other person decide when to stop. One time though, he very nearly called “stop” and was quite relieved when the other man did. He had no idea how long they had sparred, but finally he could not even hold his arms up to defend himself. He let his arms drop to his sides, and still felt that he might be able to deliver some kind of blow by swinging up. Right then, the other man said, “stop” and Moore found that a considerably pleasant sound. Most of the men, who Moore fought, were larger than he. One, in particular, was about 6’3” tall and, accordingly, had a substantial advantage in reach. This man would invariably want to “go easy,” but in the midst of sparring, either man might say “go hard” and immediately change the level of aggression. However, this particular man would decide, without announcing it, to “go hard.” Moore learned after awhile, to just “go hard” with this man. Moore’s recovery time from his bout with malaria must not have im- peded his boxing at Princeton. At one time, not long after he had begun boxing there, Spider told him, “Well, I’ll tell you what, Mr. Moore, you don’t ever have to worry about having any bad effects from malaria. I’d hate to see you if you were a totally healthy person. I’ve never seen anyone who fights the way you fight.”4 Moore would literally run over to the gymnasium, when he got out of class at Princeton, so that he would have as much time boxing as possible. Once at the gymnasium, Moore would sit and wait until someone came in who wished to box. They would box awhile, until the opponent would decide to leave, and then, Moore would sit down to wait until someone else wished to box. After awhile that opponent would leave and Moore would again sit down to wait for someone else to come along. This would go on for two or three hours. At one point he was practicing a right cross. He wanted to perfect that blow and he was boxing with a man ho was not as capable a boxer as he. Although he had no desire to hurt his opponent, the man led with a left jab only to have Moore return a right cross over the jab, more sharply than Moore had intended. The novice went down, surprising Moore perhaps even more than his opponent. In any case, Moore apologized profusely, not meaning to have hurt the man.

4Personal conversation with R. L. Moore, Austin, Texas. 68 CHAPTER 3. THE FORMATIVE YEARS

In 1931-32, Moore was the first native American mathematician to tour on the American Mathematical Society Visiting Lectureship. On that program he visited Princeton and quickly made his way to see “Spider” Kelley. Moore had been perplexed for years at an incident which had occurred while he was boxing at Princeton. He had been boxing with a man named Red Carpenter (the man had bushy red hair) and Spider just stopped them. Moore had never understood why they had been asked to stop. They were just boxing and he thought nothing was wrong. In any case, Spider had said, “Stop!,” and they had stopped. No explanation was ever given. Thus, as he found himself back at Princeton he went to see Spider. He walked into the gym as if he had not been gone at all and Spider looked at him and said, “Well, Mr. Moore, I’m surprised to see you here.” Moore replied, “Why did you stop Red Carpenter and me from fighting that time?” Spider began to tell him only to be interrupted by someone else who had come up to them. The explanation was never finished and Moore never learned why Spider had stopped them. Moore impressed others besides his sparring partners with his aggres- siveness. Once Moore’s officemate was overheard asking another faculty member, “Have you ever seen Moore go into action?” The other replied, “Well, once he grabbed a cane from my hand and nearly broke my thumb!” Moore’s officemate then related that, “I was sitting in my office talking with Moore and began talking in a way I knew I shouldn’t, and suddenly, Moore got up, I found myself on the floor, and Moore was walking out the door.” Moore had wished to express disapproval of the manner in which the officemate was talking, but did not wish to exchange words or physically fight his officemate. So he just got up, picked the man up from his chair, and dropped him sprawling on the floor. By leaving, his protest as effective and could almost be assumed to be humorous. In any event, his officemate tended to watch his tongue more closely after that. It is obvious that R. L. Moore matured much during those years immediately following his graduate study. He suffered poor health while at Tennessee, made little mathematical contributions while there, and left after only one year. Moving the next year to Princeton, he discovered the sport of boxing and seriously participated in it, both for reasons of health and because he fiercely loved the sort of aggressive individual competition boxing offered. Finding himself teaching in a setting in which he found little in common with teaching philosophies of others on the faculty, he produced little mathematics but developed strong physical and mental independence. Perhaps it is proper to say that his already strong independence of will was further developed while at Princeton. The only paper to appear, as having been submitted by Moore while at Princeton, was entitled “Geometry-on which the sum of the angles of 69 every triangle is two right angles” (Trans. Amer. Math. Soc., V. 8, 1907, p. 369-378). However, this paper had been presented to the Society on April 22, 1905, as part of a paper entitled “Sets of metrical hypotheses for geometry.” Within this paper a footnoted comment illustrates Moore’s concern for clarity of statements: “Dem” means an indication of a demonstration. Like wise, “by theorem” does not necessarily mean that this is the only theorem used in the demonstration. Similar statements may be made about the use, in this paper, of the words “therefore,” “hence,” etc. The sign = means “is, or are, identical with.” Denoting by S the assumption that the sum of the angles of every triangle is two right angles, Moore offered the following statement regarding the results within that paper: One might state this a little more suggestively, if less accurately, as follows: “While the parallel postulate, III, and thus all of that part of Hilbertian Geometry which follows, without use of his ’Axioms of ARCHIMEDES’ and ’Vollstandigkeit Axiom,’ can not indeed be proved from his other postulates I, II, IV, with III replaced by S, nevertheless it can be shown that a space concerning which these postulates (I, II, IV, S) are valid must be, if not the whole, then at least a part, of a space in which III also is true.” He further stated: This result has an interesting connection with our spatial experience. Statements have been made to the effect that, since no human instruments, however, delicate, can measure exactly enough to decide in every conceivable case whether the sum of the angles of a triangle is equal to two right angles (unless the difference between this sum and two right angles should exceed a certain minimum amount), it is therefore impossible to settle the question whether our space is Bolyai-Lobachevskian or Euclidean even though it be granted that it is one or the other. While at Princeton, Moore taught (11) Plane Trigonometry, (13) Spherical Trigonometry and Applications of Trigonometry, (15, 16) Selected Por- tions of Algebra and Elementary Theory of Equations and Conic Sections, Treated from the Cartesian Standpoint, all precalculus courses. Addition- ally, he offered a course (110) Foundations of Geometry. The description of the course, as it appeared in the 1907-1908 catalog is as follows: 70 CHAPTER 3. THE FORMATIVE YEARS

A consideration of various sets of postulates for arithmetic and geometry, and a study of relations existing between fundamental propositions. The development of arithmetic and geometry from a set of postulates. Graduate course, second term, 3 hours a week. Dr. Moore.

Moore’s strong feeling concerning the teaching methods employed at Princeton is indicated by an incident which occurred at the University of Texas over forty years after Moore left Princeton. Professor Lefschetz was an invited speaker at the University of Texas and had for many years been on the Princeton faculty, though he came there following Moore’s departure. Moore normally did not attend such lectures but, to show courtesy to a Princeton mathematician of high reputation, Moore did attend Lefschetz’ lecture, taking pains though to sit toward the back of the room. At the end of Lefschetz’ talk, as questions were being raised. Lefschetz asked Moore’s opinion of his comments. Moore had not gone to the talk seeking an argument, but as he later put it, Lefschetz looked up at him and said, “Well, do you agree with everything that I have said, Dr. Moore?” Moore, realizing that Lefschetz was not about to let him be at peace, stood up and asked, “Do you still teach algebra at Princeton the way you did when I was there?” When Moore left Princeton at the end of the academic year 1907-1908, he moved to an instructorship at Northwestern University. He remained there for three years. In 1908 the mathematics faculty numbered seven, and stressed geometry at the advanced level. In 1908-1909, Moore taught plane trigonometry and analytical geometry, algebra, solid geometry and plane trigonometry, differential and integral calculus, and was scheduled to teach ordinary differential equations though that course was not offered in 1908- 1909. This course assignment was repeated the next year, including the offering of ordinary differential equations. However, by 1910- 1911 a new course had been added, “Non- Euclidean geometry,” taught by Moore on a “to be arranged basis.” With the departure of Moore from the faculty the next year, so departed that non-Euclidean offering from the mathematics catalog. Moore did find his way back to Texas during his stay at Northwestern, at least to the extent that he found his way to Brenham, Texas on August 19, 1910 to be married to Margaret McClellan Key. The ceremony was performed at her family home with a Dallas pastor officiating. Thus, when R. L. Moore moved from Northwestern University to the University of Pennsylvania in 1911, he brought to Pennsylvania his Texas bride, but no additional mathematics in published form. It soon was to become apparent, though, that much work had been done after his gradua- 71 tion from the University of Chicago, for Moore began to earn a considerable reputation for himself as a research mathematician. The mathematician hired by the University of Pennsylvania named Robert Lee Moore was not yet thirty years of age. He had a sound academic reputation, though he had not blossomed forth with immediate substan- tive mathematics following his graduation from the University of Chicago. However, with the exception of Princeton, the University of Pennsylvania offered Moore a setting in which to function which was the most stable and substantial that he had found since leaving Chicago. The chairman of the department at Pennsylvania was named Fisher who was for many years of some reputation because of a calculus text which he authored. Those of professor rank included , Schwatt, and Hallett. The faculty at Pennsylvania was not without strength, in mathematics as well as individual strength of personality. Foremost among those who were of colorful personalities was Schwatt, an analyst who specialized in real variables and infinite series. Schwatt was not always kind in his remarks to students. As one of them recalls, “Schwatt kept saying ’Don’t you understand? Mr. Smith, there’s three kinds of fools (with a heavy accent) . . . there’s just ordinary fools, there’s damn fools, and then, there’s you.”’

Schwatt was of short stature, with a long nose, and a beard which students considered funny. He was a proselytizer of students and sought others to share his enthusiasm for series, particularly divergent series. He would chew tobacco (and spit) in the classroom. Occasionally this would be met with protest from a female student who would have backbone enough and a fracas would result. At one point Schwatt had the son of Bell, the Chairman of the Board of Trustees of the University of Pennsylvania, in one of his undergraduate classes. Schwatt, in reprimanding Bell’s son, is reported to have said, “Mr. Bell, there are fools, and damn fools and godamned fools. You belong to the third category!”5

Along with Schwatt and Fisher, Pennsylvania also had as professors and Hallett. specialized in analytic and differential geometry and Hallett was an algebraist. Additionally, in 1911-1912 there were five other faculty holding rank assistant professor. The course offerings were broad, covering analysis, algebra, differential equations, geometry, and applied mathematics. By 1912 the chairmanship of the department had shifted to , with Professors Fisher, Schwatt, and Hallett, and Assistant Professors Safford,

5Personal conversation with Joseph Thomas, Durham, North Carolina, July 18, 1971. 72 CHAPTER 3. THE FORMATIVE YEARS

Glenn, Chambers, and Babb. Instructors were Mitchell, Moore, and Beal. Moore offered a course entitled Foundation of Mathematics, described in the catalog as: The theory of positive integers as a basis for analysis. Rigid motion and correspondence with a number manifold as factors in determining the properties of space. Metrical and non-metrical spaces. A critical study of interrelations between different systems of axioms. By the following year Moore had introduced a course entitled Theory of Points Sets: Theory of sets of points in metrical and in non-metrical spaces. Contributions of Frechet and others to the foundations of point set theory. Content and measure. Jordan curve theory and other applications. These two courses Moore offered throughout his tenure at the University of Pennsylvania, usually offering each course every other year, alternating the two of them. R. L. Moore was first appointed as Instructor of Mathematics and he was to hold that rank until 1916. The mathematical period of dormancy, though, had passed for R. L. Moore. He was to have five of his papers appear in published form during the years 1912 through 1915, and two others were published in 1916. His work during that period dealt with analysis as well as continue effort toward Veblen’s axioms for geometry: Indeed, in 1912 he published “A note concerning Veblen’s axioms for geometry” (Trans. Amer. Math. Soc. 13 (1912) 74-76) and “On Duhamel’s theorem” (Ann. of Math. 13 (1912) 161-166). The first had been presented to the Society, in somewhat different form, on October 26, 1907 so it is clear that the mathematics in that paper had been completed before his move to Pennsylvania. Moore had offered some improvement of Veblen’s axioms by offering axioms alternative to Veblen’s, thereby reducing Veblen’s axioms by one. In the paper entitled “On Duhamel’s theorem,” R. L. Moore dealt with a problem of analysis, using analytic methods. Duhamel had posed the proposition:

Let αl + α2 + ... + α be a sum of positive infinitesimals which approaches a limit when n = . Let βl +β2 +...+β be a second sum of infinitesimals which differ∞ from the infinitesimals of the first sum by infinitesimals of higher order; i.e., let α lim i = 1. βi 73

Then the second sum approaches a limit when n = , and this limit is the same as that of the ﬁrst sum:

6 lim βl + β2 + ... + β = lim αl + α2 + ... + α. n= n= ∞ ∞

Osgood had formulated an alternative theorem which was without some of the difficulties of application encountered by use of Duhmel’s theorem. Then Moore proceeded to offer another version of the theorem “which is suitable for application to a still wider range of problems and at the same time seems to be even easier to apply.”7 By 1915 R. L. Moore was devoting more attention to “the linear continuum.” On April 24, 1915 he presented a paper “On the linear continuum” to the American Mathematical Society, in which he refers to a set of eight axioms which he had earlier proposed (The linear continuum inG terms of point and limit, Ann. of Math. s. 2, v. 16, 1915, p. 123-133) for the linear continuum in terms of “point” and “limit.” In that paper he had defined “betweenness” and had established that the set is categorical with respect to “point” and the implied notion of “betweenness.”G He offered a new axiom, replacing Axiom 5 of the original set so as to gain a system which was “absolutely categorical,” in the sense of Veblen (A system of axioms for geometry, Trans. Amer. Math. Soc. v. 5, 1904, p. 343-384). It was in 1915 that R. L. Moore published a result which gained him the attention of the mathematical community. Moore and Veblen had both treated geometric problems in their dissertations. However, Veblen’s work seemed to be taking him toward a notion of non-metric considerations while Moore’s work seemed to be taking direction toward the metric situation. In fact, Veblen had published in 1905 a paper entitled “Theory on plane curves in non-metrical analysis situs” (Trans. Amer. Math. Soc. 6 (1905) 83-98) in which he argued a proof of the fundamental theorem proposed by Jordan: A simple closed curve lying wholly in a plane decomposes the plane into an inside and an outside region. The setting for this theorem was taken to be a space satisfying Axioms I - VIII, XI of his dissertation, stating explicitly that nothing is assumed “about analytic geometry, the parallel axiom, congruence relations, nor the existence of points outside a plane.” The results published in 1915 by Moore were rather devastating, inasmuch as he established that any plane satisfying Veblen’s Axioms I - VIII, XI is a number plane. That is, any plane, satisfying those axioms, contains a system of continuous curves such that, considering those curves as straight lines, the plane is an ordinary Euclidean plane. Thus, Moore

6Annal. Math. 13 (1911-12) 161-166. 7Ann. of Math. 13 (1911-12) 161-166. 74 CHAPTER 3. THE FORMATIVE YEARS claimed that any discussions of analysis situs based on Veblen’s axioms (as for example, Veblen’s proof of the theorem that a Jordan curve divides its plane into just two parts) is no more general than one based on analytic hypothesis. The results obtained by Moore were remarkable and unexpected by many. The mathematics involved was diﬃcult, in that Moore established a one-to-one reciprocal continuous correspondence between the continuous curves of Veblen’s planes and straight line of an ordinary Euclidean plane. If Robert Lee Moore had not a substantial reputation before publishing those results, he surely had established himself after those result were known. Moore’s attention to and interest in geometry and particularly Veblen’s axiomatic approach had been consistent across the years since he departed from Chicago. In 1913 he presented a paper before the American Mathe- matical Society, “Concerning a non-metrical pseudo-Archimedean axiom,” in which he had further investigated Hilbert’s plane axioms of Groups I and II8 and Veblen’s Axioms I - VIII. It was after his remarkable result which showed Veblen’s axioms really gave rise to a metric plane, that Moore began to oﬀer theorems which were not truly set in analysis nor were they straightforward investigations of axioms of geometry. In his paper, “On the foundations of plane analysis situs,” (Proc. Nat. Acad. Sci. 2, 1916, pp. 270-272) he put forth the fundamental notions which were to form a basis for later development in point set topology. Indeed, in that paper, Moore stated:

The notion point, line, plane, order, and congruence are fundamental in Euclidean geometry. Point, line and order (on a line) are fundamental in descriptive geometry. Point, limit-point and regions (of certain types) are fundamental in analysis situs. It seems desirable that each of these doctrines should be founded on (developed from) a set of postulates (axioms) concerning notions that are fundamental for that particular doctrine. Eu- clidean geometry and descriptive geometry have been so developed. The present paper contains two systems of axioms, Σ2 and, Σ3 each of which is suﬃcient for a considerable body of theorems in the domain of plane analysis situs. The axioms of each system are stated in terms of a class, S, of elements called points and a class of sub-classes of S called regions.

Within that paper he stated the following deﬁnitions and axioms. These were to be the forerunners of remarkable developments in point set topology for at least the next half century.

8Bull. Amer. Math. Soc. 22 (1915-16) 225-236. 75

Deﬁnitions

A point P is said to be a limit point of a point-set M if, • and only if, every region that contains P contains at least one point of M distinct from P. The boundary of a point-set M is the set of of all points • [X] such that every region that contains X contains at least one point of M and at least one point that does not belong to M.

If M is a set of points, M 0 denotes the point-set composed • of M plus its boundary. A set of points K is said to be bounded if there exists a ﬁnite • set of regions R1,R2,R3 ...Rn such that K is a subset of (R1 + R2 + R3 + ... + Rn)’.

If R is a region the point-set S R0 is called the exterior • of R. − A set of points is said to be connected if however it be • divided into two mutually exclusive sub-sets, one of them contains a limit point of the other one.

The System Σ2

Axiom 1. There exists an inﬁnite sequence of regions, K1,K2,K3 ... such that (a) if m is an integer and P is a point, there exists an integer n greater than m, such that Kn contains P , (b) if P and P are distinct points of a region R, then there exists an integer δ such that if n > δ and Kn contains P then K0 is a subset of R P . − Axiom 2. Every region is a connected set of points.

Axiom 3. If R is a region, S R0 is a connected set of points. − Axiom 4. Every inﬁnite set of points lying in a region has at least one limit point. Axiom 5. There exists an inﬁnite set of points that has no limit point. Axiom 6’. If R is a region and AB is an arc such that AB A is a subset of R then (R+A) AB is a connected set of− points. − Axiom 7’. Every boundary point of a region is a limit point of the exterior of that region. 76 CHAPTER 3. THE FORMATIVE YEARS

Axiom 8. Every simple closed curve is the boundary of at least one region.

The System Σ3

The system Σ3 is composed of Axioms 1’, 2’, 3, 4’, 5, 6’, 7,’ and 8, where Axioms 1’, 2’, and 4’ are as follows:

Axiom 1’. If P is a point, there exists an inﬁnite sequence of regions, R1,R2,R3 ... such that (a) P is the only point they have in common,

(b) for every n Rn+1 is a proper subset of Rn, (c) if R is a region containing P then there exists n such that R0 is a subset of T . Axiom 2’. If A and B are two distinct points of a region R then there exists, in R, at least one simple continuous arc from A to B.

Axiom 4’. If R is a region, R0 possesses the Heine-Borel property.

An example of a system satisfying Σ2 is obtained if in ordinary Euclidean space of two dimensions, the term region is applied to every bounded connected set of points M, of connected exterior, such that every point of M is in the interior of some triangle that lies wholly in M.

At about this time, on April 6, 1915, Dean Arthur Hobson Quinn wrote, as chairman of the College of Arts and Sciences nominating committee, to Dr. Josiah H. Penniman, Vice-Provost, “The Department of Mathematics recommends the promotion of Dr. Robert Lee Moore, Instructor in Math- ematics, to the position of Assistant Professor of Mathematics . . . ” On May 4, 1915 Dean Quinn wrote Dr. Edgar F. Smith, Provost, a letter in which the same statement was made. The promotion was approved and the academic year 1916-1917 saw Robert Lee Moore at the University of Pennsylvania holding rank Assistant Professor. The year 1916 was one in which R. L. Moore graduated his ﬁrst doctoral student. J. R. Kline graduated that year, having ﬁrst come to the University of Pennsylvania in 1912 after graduation from Muhlenberg College with an A. B. degree. He received his M. A. from Pennsylvania in 1914 and his Ph.D. in 1916. His formal courses, with fellow students, under Moore, were not many. Moore alternated from one year to the next his courses 77 entitled “Foundations of Mathematics,” and “Theory of Sets.” Beyond those, individual study was the fashion, with Moore encouraging some to work with him while discouraging others. Kline was able to establish the following converse to the Jordan theorem:

Suppose K is a closed set of points and that S K = S1 + S2, − where S1 and S2 are non-compact point sets such that

1. every two points of Si (i = 1, 2) can be joined by an arc lying entirely in Si 2. every arc joining a point of S to a point of S contains a point of K 3. if 0 is a point of K and P is a point not belonging to K, then P can be joined to 0 by an arc having no points except 0 common with K. Every point set K that satisﬁes these conditions is an open curve.9

Kline was to prove to be a substantial member of the mathematical community and was, in fact, to spend most of his career at the University of Pennsylvania, returning there in 1920 and remaining there until his death in 1955. By 1918 Moore had further developed his mathematics and his teaching techniques. Graduating in 1918 was G. H. Hallett, a son of Professor Hallett of the mathematics faculty. An idea of the manner of teaching used by R. L. Moore is given by Hallett’s description:

He taught in a very remarkable way. He didn’t give us any books. We didn’t consult books at all in that course. It was a course in point set theory and he gave us certain axioms to start with and then we were asked at the beginning to prove certain theorems that we were told were true, given those axioms. We would work on the proofs and come back into class and he would ask how many people had the proofs and those who said yes were given a chance, at least one or two of them to give their proofs. And the other members of the class listened carefully to see if they made any mistakes and if a member of the class thought so, he would speak up and say why. And quite often, there were mistakes in the proof that were caught by the class and sometimes, nobody had the proof the ﬁrst time and he would let us have another week at that one. Ordinarily, we would

9The converse of the theorem concerning the division of a plane by an open curve. Trans. Amer. Math. Soc., J. R. Kline, XVII (1917) 178. 78 CHAPTER 3. THE FORMATIVE YEARS

come in with the proofs. As the course went on though, some of the things given us were more diﬃcult. He would give us a problem and ask what the solution to the problem was without telling us the theorem that was supposed to result. Then we would work on that. I remember that one of the theorems that I proved, he said had never been proved until the year before, and I had given him a diﬀerent proof. I think, I’m not sure, he may have published that, but that’s one case of that kind that I remember. He gave us a problem once which we worked on and which none of us got. He gave us a couple of weeks to work on it and none of us got it. And he said, “Well, I guess you needn’t spend any more time on that. This is a problem mathematicians have been working on for centuries and nobody has ever solved it. I just thought you might, just by accident, be able to do something.10

It well could be that Moore’s manner of teaching inﬂuenced others. Hallett describes the teaching style of Mitchell of the mathematics faculty, reminding one at once of Halsted’s style of teaching:

One other course I took at the same time which was somewhat similar. It was taught by Professor Mitchell whom I also liked very much. He took a book, I think it was by Dr. Pierpont at Yale. I think it was in the area of functions of a real variable. I guess Professor Mitchell had found on inspection that not all of Professor Pierpont’s proofs held up, so the way he taught this course in that subject, he gave us this book, but asked us to go through all the proofs that were given and find out whether they were watertight proofs or not and if not, why not. So time after time, we would find that they weren’t and it almost always occurred when the author said, ”Now, it is evident that . . . “ And we would inspect and we would find quite often that it wasn’t even so. It seemed plausible that it wasn’t so. Actually this course had many of the elements of the other course. We’d discuss the matters in class together11

Moore also had incorporated some of Halsted’s teaching style in his own approach to the classroom activity. Halsted often had been known to

10Personal conversation with George Hallett, New York City, New York, October 8, 1971. 11Personal conversation with George Hallett, New York City, New York, October 8, 1971. 79 discuss matters foreign to mathematics in his classes. It was toward the end of World War I that Hallett had a course with Moore and recalls:

. . . that was during the last days of the first world war and Dr. Moore would frequently come in the class and take five minutes discussing something that had just appeared in the papers about the arguments of the logic, or lack of logic, in some statements by an important public figure connected with the war or other public events at that time. He and I were always getting a good deal of satisfaction in picking out the lack of logic in a good many of these statements by important public figures. That was sort of a forerunner to applying the same methods of thinking to public affairs which I went on with after graduation and for the rest of my areer. I think this added to the interest of the class and probably made it more attractive. And I think it was really of great value to point out that the same sort of criteria could be applied widely in fields where we weren’t just dealing with mathematics.

Hallett continued:

I think I should make the observation that Dr. Moore’s method of teaching brought out what appeared to me to be the two most important faculties in mathematical research - one which would surprise most people, I’d regard as imagination and the second, the ability to critical analysis in applying logic to what you think of to try out. And this same criteria, of course, apply to almost everything else. It is a method of thinking. And I think such success as I’ve had in the work I’ve done in the ﬁeld of government probably has a good deal to do with that - because they don’t catch me up very often in theories of logic in bills, or diﬀerent parts of bills, that don’t hang together.12

Hallett, upon graduation, was oﬀered a job as a secretary of the Pro- portional Representation League. He rejected an oﬀer of a position at Rice Institute to accept a post with the League and went on to an illustrious career in public service. While Moore was at the University of Pennsylvania the Department of Mathematics was housed in College Hall, the original building in West Philadelphia for the University of Pennsylvania. It was a stone building

12Personal conversation with George Hallett, New York City, New York, October 8, 1971. 80 CHAPTER 3. THE FORMATIVE YEARS and Moore was oﬃced in the basement in a fairly large room, with seven or eight faculty of instructor or assistant professor rank. It was right next door to a stable where they kept the horse that pulled the lawn mower over the campus. It was not a very pleasant place for faculty oﬃces. At least one of Moore’s students from Pennsylvania likened him to a football coach, ”He was always recruiting people.”13 One of the students he recruited was Anna Mullikin, who had come to the University of Penn- sylvania from Goucher College. Moore did not hesitate to recruit a female student, so he quickly introduced her to his style of teaching. In some classes other students felt the classroom activity omitted them, with most of the conversation occurring between Moore and Mullikin. Anna Mul- likin’s experience with Moore was similar to that of Hallett. She, too, was impressed with his unusual method, recalling:

He had his work all published, you see, and people would go and look it up in the library and he didn’t want them to do that. He wanted them to work it out themselves. And he put them out of class if he found out that they were cheating. (In one class) there were three of us, but he put everybody out but me. One was a Catholic nun and she tried to get help from me and he put her out. He said if she needed help she didn’t belong in his class.14

By the post World War I years, Moore had developed his fundamental approach to teaching. He was to modify it, depending on the setting and the level of students, but the foundations of the method were not to alter. He would pose problems for the students and demand that they attack those problems on their own initiative and ability, resorting not at all to source books or other people. He was with extreme patience toward those students who would struggle through eﬀorts at proofs, but with extreme impatience toward those who would violate the rules of study which he laid before them. He simply and immediately dismissed them from class. Though he found support for his approach among some, there were those students who would not tolerate such a style and faculty who opposed such a method. Moore’s method of teaching graduate mathematics oﬀered contrast with the catalog description of requirements for a Ph.D. in 1913- 1914:

Students were especially advised not to undertake an unduly large number of lecture courses at one time. The place for grad-

13Personal conversation with Joseph Thomas, Durham, North Carolina, July 18, 1971. 14Personal conversation with Anna Mullikin, Philadelphia, Pennsylvania, July 24, 1971. 81

uate study is the library rather than the lecture room. The most that can be done in the lecture is to guide the student into a general acquaintance with the principles of a subject, and it remains for him to broaden his knowledge and develop the details by extensive reading and private study.

Moore’s personal values were not to be tampered with. He felt strongly about his southern heritage, honor, and gentlemanliness. One of his office mates was named Beal, a pleasant person who was slightly crippled. He and Moore were good friends, often lunching together. Moore seemed to some, including Beal, to prefer having girls in his classes. Perhaps he felt they were easier to draw out. Once Beel began kidding Moore over his preference for female students and Moore took offense. He didn’t wish to hit Beal, but wished to make his protest effective. So, he quickly arose, picked Beal up from his chair and dropped him on the floor as he left the room. By taking that tack, the incident could be considered as humorous, but the point was well made. Later, someone reportedly asked Moore, “Well, what about that? Why did you do such a thing?” And Moore said, “Well, I couldn’t hit a cripple!”15 Moore did not successfully recruit all those students he sought. One young lady, Clara Williamson, had gained entry into the University of Penn- sylvania, though it was predominantly a male school. She took calculus and differential equations from Moore and then was encouraged by Moore to study for her doctorate with him. She decided against such an effort, even though he had invited her to work toward her Ph.D. with him. While Anna Mullikin was involved in her dissertation effort, in 1919, Moore came into his office in College Hall in a state of some excitement and sat down in a chair by the desk of a young instructor, Joseph Thomas. The only telephone in the room was on that desk and Moore evidently had something important on his mind. Thomas offered to leave, being the only other person in the office at that time. Moore said, “Oh, no, no, it doesn’t make any difference,” and he called , the chairman of the department. Moore informed that he had received an offer of an Associate Professorship from the University of Texas and made an appointment to see about it. The outcome was that R. L. Moore chose to return to Texas, accepting the Associate Professorship at his alma mater. At that time, he was an established mature mathematician, who was beginning to have success in producing students. He was 37 years of age and had been at Pennsylvania for almost ten years. He had one doctoral student in progress and she was to follow him to Texas and complete her dissertation there.

15Personal conversation with Joseph Thomas, Durham, North Carolina, July 18, 1971. 82 CHAPTER 3. THE FORMATIVE YEARS

Halsted had been gone from Texas for years, having left there in 1902. But M. B. Porter, Halsted’s student who had achieved early success and who was on the faculty at Texas while Moore was an undergraduate, had gone away and now had returned to the University of Texas as departmental chairman. The idea of returning to Texas had natural appeal for Moore, as it often has for many Texans who find themselves on “foreign soil.” He had some knowledge of Porter and an idea as to how he could function in a department chaired by Porter. Then, too, Mrs. Moore had not been in good health. She had suffered a severe fall in a subway and Moore had been careful to nurture her health. Perhaps the warmer climate of Texas was inviting to them both. In any case, the academic year 1920-1921 found R. L. Moore no longer at the University of Pennsylvania. During Moore’s tenure at Pennsylvania, he evolved from a promising mathematician to one of stature. He began serving as an Associate Edi- tor of the Transactions of the American Mathematical Society in 1913 and continued in that capacity until he left Pennsylvania. He was a member of the Council of the American Mathematical Society in 1917-1919. He had produced substantial mathematics during his years at Pennsylvania and had begun the developments away from pure geometry toward those concepts which would become known as point set topology. He produced two doctoral students prior to leaving Pennsylvania (Kline and Hallett) and had another in progress (Mullikin) who would formally receive her degree from the University of Pennsylvania, though completing her dissertation while at Texas. Of his three doctoral students, each would make substantial contribution to his chosen profession, though the three would work in decidely different settings. Moreover, each would credit Moore’s influence even though his student-teacher contact was brief. Kline would establish himself well as a university academician. Mullikin would have a long, worthwhile career as a high school teacher of mathematics. Hallett would experience an illustrious career in public service. Each would feel that Moore’s style of teaching gave far more than just information to them; it developed their power of rational thought and this would stand them in good stead throughout their careers. The University of Texas, with the hiring of R. L. Moore, had added to its faculty an established teacher, researcher, and producer of doctoral graduates. At the time of his first appointment, his research already accomplished gave greatest support to his professional reputation. He was gaining acclaim as a teacher. In fact, he had influenced E. H. Moore to teach in the manner which R. L. Moore had developed. While visiting at the Uni- versity of Chicago one summer, he had lunched one day with E. H. Moore and L. E. Dickson. A topic of the conversation then between them dealt with effective methods of teaching mathematics. R. L. Moore explained 83 his method, describing the procedure of posing questions or theorems for students and insisting that the students settle those questions on their own, without assistance of any sort save their own capability. Dickson tended to quickly deride that approach, but E. H. Moore, as was his wont, said little. He customarily gave some thought to new ideas before reacting to them. Later, E. H. Moore told R. L. Moore that he felt that approach was with much merit, and began employing it in his own classes. Even though he was known as a capable researcher and gaining stature as a teacher, perhaps no one at all predicted the extreme success which was to be R. L. Moore’s at Texas. It was to be almost as if he had left Texas, spent time wandering about preparing himself until both he and his method were seasoned and mature. Then he returned to Texas to wring remarkable results from a raw and rather undeveloped setting. 84 CHAPTER 3. THE FORMATIVE YEARS Chapter 4

Return to Texas

The University of Texas to which R. L. Moore returned in 1920 was much different from the institution he departed in 1902. Halsted had departed Texas some years earlier, remaining there only one year after R. L. Moore graduated. The Department of Mathematics fell into a period of dormancy following the departure of Halsted. Halsted’s enthusiasm and drive were not picked up by any other member of that faculty. Porter, Halsted’s first protege to have left Texas to gain a doctorate elsewhere, and who had returned to teach at Texas while R. L. Moore was an undergraduate student there, had departed since, only to return again and now was the only mathematics faculty holding Professor rank during the academic year 1919-1920. Associate Professor E. L. Dodd, a specialist in actuarial mathematics was chairman of the department in 1919-1920. In addition to Porter and Dodd, others on the mathematics faculty in 1919-1920 included Adjunct Profes- sor Ettlinger (calculus of variations and differential equations), Associate Professor Calhoun, and Instructors Decherd, Horton, Batchelder, Burnam, and Roberson. Instructor Decherd was the young woman who, some seventeen years earlier, had been hired as a tutor, instead of R. L. Moore, and over Hal- sted’s strong objections. Halsted probably would have appreciated the course exchange which occurred on Moore’s return to the Texas faculty. Miss Decherd had been scheduled to teach a course entitled Foundations of Geometry. A student of that year recalls:

. . . it was scheduled for Miss Decherd and she was supposed to teach it and I got in her class and she didn’t show up, and Dr. Moore did. That’s what happened. And he changed the course all around. It was supposed to be a geometry course

85 86 CHAPTER 4. RETURN TO TEXAS

and it just opened up a whole new world for me. I had never been taught like that. He came in and he wrote some deﬁnitions on the board and then he talked to us awhile and then gave us some axioms. Then he gave us some theorems to prove. He kept giving theorems and there were only about ﬁve in the class. I really had a ball. He wrote some theorems on the board and then he would bring up some questions. “Do you think this would be true about it?” and “Do you think that would be true?” When you decided whether or not it would be true, “Well, can you prove it?” And when you said you had a proof, he would ask you to go prove it. And then, if there was someone in his class who thought he was pretty smart and tried to use mathematical terms, maybe and try to pretend he knew more than he did, Dr. Moore really would pick on him.1

The student, Blanche Bennett, spent her own professional career teaching at the university and college level, though she never earned her doctorate. She did earn her Master’s degree at the University of Texas, one of only four who Moore directed at that level. Moore had wished her to continue working toward her doctorate and was disappointed when she failed to do so. His unusual time in thought and consideration devoted to students outside of class is indicated by his informing Mrs. Moore of the progress of some of them. Blanche Bennett, upon being asked whether Moore had encouraged her to continue past her Master’s with him, responded,

Yes. He was cute. I got married during the year that I earned my Master’s degree. I married just before I left Austin. I was still in his class, but when I went back I signed my name diﬀer- ently, GroverGuy,, not BennettBing, R. H. - and Mrs. Moore told me what he said. It seems as though he talked about his students with her, graduate students, anyway. Mrs. Moore said Dr. Moore kept talking about “Mrs. GroverGuy,, Mrs. Grover- Guy,,” . . . and Mrs. Moore said, “Well, Mrs. GroverGuy,. That’s a new one. Has she been in your class before?” “Yes,” he said. “Well, you never did mention her.” “Well, I thought I had,” and he kept her going several days before he told her about it. But when I left there, he didn’t like

About Moore’s eﬀectiveness as a teacher, Blanche Bennett said,

1Personal conversation with Mrs. Blanche Bennett Grover, Houston, Texas, Decem- ber 18, 1970. 87

One thing is his personality. He has such a forceful personality. I think he’s able to do a lot of things that most people can’t do. He’s able to put over his own teaching so much better than most people can. It requires more eﬀort. That’s not the easiest way to teach. I think the easiest way to teach is to write a proof on the board and let the student copy it and then give it back to you. I think some people do that. I know some at the university that did it. I don’t think you learn anything that way. How do you know that proof’s all right, if you don’t understand it? And I’m sure some of them didn’t.

The characteristics of Moore’s teaching were easier to identify by the time he began teaching at Texas. He placed the student on his own, demanding that no outside assistance be used, and offering the student problems or theorems which were within his capability to solve. He thought much about his students, individually, and kept Mrs. Moore well-informed as to the progress of many of them. He had begun developing his talents for recruiting better minds and was becoming a forceful collector of brilliant students. An example of R. L. Moore’s power to attract quality students to his mathematics is offered by his first doctoral student at the University of Texas, R. L. Wilder. Wilder had earned both his Bachelor’s and Master’s degrees at Brown University prior to his coming to study at the University of Texas in 1921. His attraction to Texas was not that of R. L. Moore; rather, he had decided to become an actuary and Dodd, a member of the Texas mathematics faculty had strong reputation for producing capable people in that profession. Wilder, feeling it reasonable to take other courses in mathematics while pursuing his own interests, enrolled in a course under Moore. He shortly became captivated by the mathematics Moore offered and saw his own talents flourishing in that setting. He graduated with his doctorate in 1924, dealing in his doctoral thesis with continuous curves, gaining a sufficient condition that a continuum be a continuous curve. Moore earlier had established that the condition was necessary, so the characterization thus offered was complete. Wilder went on to have an unusually effective career as a mathematician, authoring three books and over seventy papers, holding several offices in highly regarded academic societies, and was elected to membership in the National Academy of Sciences in 1963. Society may have lost a fine actuary but it surely gained a superior mathematician. Another premier example of Moore’s capability as a teacher and re- cruiter is given by the example of his third doctoral student at Texas, G. T. Whyburn. Whyburn had come to the University of Texas from a small community in Denton County, north of Dallas. He came to study chem- 88 CHAPTER 4. RETURN TO TEXAS istry, but happened to take a calculus course in which R. L. Moore was the instructor. Moore recognized Whyburn’s potential in mathematics and continued to encourage him to give up chemistry for mathematics. Why- burn continued taking courses from Moore, while proceeding as a chemistry major. He accepted his Master’s degree in chemistry and Moore kept telling him that he had most unusual ability in mathematics and that it was a shame for him to go into chemistry, his ability in mathematics was so great. Finally, Whyburn decided to work in mathematics and one year later earned his doctorate. Nearly all his mathematics had been taken with Moore and had been, for the most part, during the time he was actually pursuing a degree in chemistry. Thus, upon graduation with his doctorate in mathematics, he was a product of Moore’s teaching. There would arise, across the following years, an argument against any student being allowed to graduate after having studied under only one man. Against that argument were offered examples of eminently successful products. Whyburn proved to be one of them. In 1957 G. T. Whyburn was elected to membership in the National Academy of Sciences. He was the first student who worked under Moore’s direction to gain that honor. Though Moore’s style of teaching was well defined as he came to Texas, his access to students was to undergo quite an evolution, across the years of his service there. In the 1920’s Moore really had no sequence of courses which were “his” and from which he could predictably find talented students. Instead, as with others on the faculty, he taught a variety of undergraduate courses, offering the student opportunity to specialize with him at the graduate level. Moore’s second doctoral student at Texas, R. G. Lubben, offers an example of a student who first encountered Moore at the graduate level. Although Lubben took all of his undergraduate work at Texas, he first studied with Moore taking a course entitled Point Sets and Continuous Transformations, described in the catalog as “A critical study of point-sets, curves, regions, etc. Senior or graduate standing and consent of the instructor.” By 1924 Moore offered another graduate course in mathematics, Foundations of Mathematics, described thusly: “A critical study of the foundations of mathematics. Senior or graduate standing and consent of the instructor.” In addition to the two courses at the graduate level, Moore regularly taught a course entitled Introduction to the Foundations of Geometry, as well as advanced calculus, calculus, and courses prerequisite to calculus. All such courses were offered in the Department of Pure Mathematics. Students majoring in physics and chemistry traditionally took their mathematics within the Department of Pure Mathematics. The Department of Mathematics, during Moore’s early years at the University of Texas, was housed in the auditorium of the Old Main Building. 89

(The Old Main Building no longer stands and the new Main Building stands on the same site.) The auditorium had been condemned and on each side of the auditorium partitions had been erected for offices. At first Moore was officed in the first office on the left as one entered the left-most door of the auditorium. Later he moved down that side of the auditorium to the other end of that row of partitions, so that he was down near the old stage. Ettlinger and Vandiver were officed across the auditorium, with Lubben and Dodd officed on the same side of the auditorium as Moore. There were scattered offices for student assistants and also some small cubbyholes generally available for the teaching assistants to deposit books and papers. A student from those years recalls:

The old auditorium served as a kind of meeting place. Now we have what we call these coﬀee rooms or common rooms where the students kind of gather around a coﬀee pot or something. Well, we gathered in the center of the old auditorium and, I suppose, not being conscious that our professors were over there, we would sit out there and discuss everything that would happen to us and sometimes the professors would come out and join us. Many’s the time we had a discussion with Dr. Moore over politics or something else - not mathematics - in the center of that auditorium with nearly half a dozen students standing around or sitting in those old sort of chairs left over from the days when they used it as an auditorium. It wasn’t what you would call elegant, but we enjoyed it.2

Moore’s habits of spending time on campus and being available to students were already well formed. Upon returning to Austin, he and Mrs. Moore rented an apartment for a short while, but after a few months they purchased a house on West 23rd, only a few short blocks from the campus. Moore would walk between his home and the campus. That house was to serve the Moores well as it was to be their only permanent residence in Austin. A description of his campus habits, as given by one of his students, testiﬁes also to his consistency. The description could have come from any of the years he was at Austin.

I believe that Dr. Moore came and stayed from nine until ﬁve or something like that. That is, from the time he came in the morning to teach, he then went home at a regular time for lunch, but he didn’t take more than about an hour or an hour and a half,

2Personal conversation with Mrs. Lucille Whyburn, Charlottesville, Virginia, March 28, 1971. 90 CHAPTER 4. RETURN TO TEXAS

and then he came back and spent the whole afternoon there. He had no classes that I can remember of in the afternoon. I don’t remember ever taking a class from him in the afternoon, but he was there in his office or out in the auditorium. And if you wanted to go and see him, he was always available. I never knew him to refuse to see a student. Now, he may have done so, but he certainly never refused to see me. His help was always given openly and freely and, I think, had a great deal to do with your feeling of comfort about mathematics, and instilling in you a desire to really understand it. So many people now talk in what I call a supercilious fashion about mathematics and as long as they can keep the conversation on a monologue, they just go along fine. They just sort of startle you with their ren- dition but then, if you just horn in on them a little, about their depth of understanding of a notion and very often you will find it’s shallow. They don’t have any depth of understanding and this is part of Dr. Moore’s real success as a teacher. He creates in students the desire for and the comprehension and the depth for a lot of understanding.3

Moore’s first decade at the University of Texas witnessed a mathematical evolution from geometry to topology, as evidenced by his research and those researches of his students. Moreover, Moore’s influence was becoming apparent as his students accepted posts of responsibility at universities elsewhere. Kline had replaced Moore at Pennsylvania, Wilder moved to the University of Michigan in 1926 after a short stay at Ohio State University, and Whyburn would move to the University of Virginia in 1934 after a brief tenure at Johns Hopkins University. There would soon develop a collection of topologists, consisting of Moore and his graduated students, who would dominate that segment of mathematics called point set topology. Indeed, a graduate under one of Moore’s students might well take his first permanent post under another of Moore’s students at another institution. Students who studied with Moore during the 1920’s, who earned their doctorate under his direction, were R. L. Wilder (1923), R. G. Lubben (1925), G. T. Whyburn (1927), J. H. Roberts (1929), C. M. Cleveland (1930), and J. L. Dorroh (1930). Of those six students, two later gained membership in the National Academy of Sciences (Wilder and Whyburn), Roberts was to have a distinguished career at Duke University, and Cleve- land and Lubben were to spend their entire career at the University of

3Personal conversation with Mrs. Lucille Whyburn, Charlottesville, Virginia, March 28, 1971. 91

Texas. Dorroh was to move through several institutions, some of high reputation, finally retiring from Texas A & I in Kingsville, Texas. By the end of the decade of the 1920’s the depression had occurred and students graduating in mathematics, seeking a university post, had not the prospect of a rich solvent career spread before them. In part, that may tend to explain why two (Lubben and Cleveland) of Moore’s first six students to graduate from the University of Texas remained there throughout their careers. Positions initially were not easy to find and, upon accepting an appointment at Texas, Lubben and Cleveland perhaps found it more to their liking to remain there than to leave. An investigation of Moore’s mathematics, as well as that of his students, establishes that by 1920 a substantial evaluation had occurred, developing an approach to investigating the plane and spaces of n-dimensions which was quite new and distinct from the geometric considerations of some ten or fifteen years earlier. An illustration of this development is offered by theorems included in Anna Mullikin’s dissertation, 4 in which she investigated connectedness of plane points sets:

Theorem 1 If, in a plane S, K and M are two closed, mutually exclusive point sets and H is a closed, bounded, connected point set having at least one point in common with each of the sets K and M, then there exists a point set H, a subset of H, such that H is connected and contains no point of either K or M, but such that K and M each contain a limit point of H.

Theorem 2 If, in a plane S, H is a closed, bounded point set containing two mutually exclusive, closed point sets K and M, but containing no closed connected, subset containing a point of K and a point of M, then it is the sum of two mutually exclusive, closed sets, of which one contains K and the other contains M.

Theorem 3 If M is the sum of a countable number of closed, mutually exclusive point sets M1,M2,M3,... no one of which disconnects a plane S, then M does not disconnect S.

During the years between 1920 and 1930 Moore became known as a leader of new mathematical developments. He was successfully generalizing, from a geometric background, and developing an approach to mathematical problems which is outlined in his paper “Foundations of Analysis Situs.” The courses which he taught were taking on a character of a theorem sequence nature, as Moore developed more theorems on the basis of his

4Certain theorems relating to plane connected point sets. Trans. Amer. Math. Soc. 24 (1922) 144-162. 92 CHAPTER 4. RETURN TO TEXAS axioms. The students would be presented with theorems which were only recently proved. They found themselves quickly on the frontier of research. Moreover, much mathematical activity was centering about Robert Lee Moore. His academic activity was visible at each level. In 1921 he presented several talks to the honorary mathematical society, Pentagram, on non- Euclidean geometry.5 That sort of activity was continued by Moore. For example, in 1933 he presented a talk on “Compact continua which contain no continuum that separates in the plane” to the newly formed Mathematics Club. He was quite often in attendance at mathematical meetings, both at the state and national level. The first summer that Moore had been at the University of Texas, J. R. Kline had been also on the faculty there, and would fill the post at the University of Pennsylvania which Moore had vacated. There would develop, across the years, an association between Moore at Texas, Kline at Pennsylvania, Whyburn at Virginia, as well as others which would allow students to move from one place to the other, either during their graduate study or following the formal completion of it. As a student would progress through certain mathematics, Moore or one of his early doctoral students would describe to another the student’s capability by naming a certain difficult theorem he had proved. As the formality of the theorem sequence developed, it would suffice to say, “He proved theorem 22” and each would know that the student was a person of accomplishment of no mean sort. Toward the end of the decade of the 1920’s greater recognition was being given to Moore and his contributions. In 1929 he was chosen by the Gradu- ate Council of the University of Texas to be a University Research Lecturer. The requirements and responsibilities of that position are indicated by:

The University of Texas has recently established what is called The University of Texas Research Lecturership with the object of encouraging research among the members of the faculty of the University and of impressing upon the minds of the students the importance of research. The lectureship is to rotate from year to year among various groups of departments in the College of Arts and Sciences. The lecturer is chosen each year by the Graduate Council of the University after a most careful investigation of the qualiﬁcations of the members of the faculties of the departments in the College in which the award is made. The holder of the lecturership is to deliver in March of

5That year C. P. Boner was president of the Pentagram. Boner, following completion of his graduate work, later assumed the post of dean of the College of Arts and Sciences, and worked toward the joining of the Department of Pure Mathematics with the Department of Applied Mathematics. 93

the year of award not less than three and not more than ﬁve lectures in a chosen ﬁeld of investigation. These lectures and other research studies of the lecturer are to be published by the University and be given publicity and distribution.6

The following year he was invited by the Council of the American Math- ematical Society to be a Visiting Lecturer for the Society. He was the first American so invited and, in fulfillment of the post, lectured at the Univer- sity of California at Berkeley, the University of Southern California, Leland Stanford, the University of Washington, the University of Minnesota, the University of Chicago, Harvard, Swarthmore, the University of Pennsylva- nia, the University of Michigan, Princeton, the University of Cincinnati, the University of Buffalo, Northwestern, the University of Iowa, the University of Wisconsin, Duke University, and Rice Institute. Those who preceded him in that assignment, as Visiting Lecturer for the American Mathematical So- ciety, included Professor Wilhelm Blaschke of the University of Hamburg, Professor Constantine Caratheodory of the University of Munich, Professor Herman Weyl of Zurich Technical School, and Professor Enrice Bompiani of the University of Rome. The end of the 1920’s brought the experiences of the Great Depres- sion. Those who went to college then, and early in the 1930’s, often did so at extreme sacrifice, or sometimes because there was nothing else to do. Promising students, with family obligations, would at times find jobs and would remove themselves from further educational efforts. Others would find such jobs temporarily and would pursue their studies intermittently. In looking backward on Moore’s career, it is easy to wonder how much greater would have been his contributions if the 1930’s had presented students in the numbers of the post World War II years. By 1932 Moore was fifty years of age; a powerful and established mathematician, in many ways his professional strength was greater than any earlier period. That teaching talent and the stimulating research mind found a student population decimated by economic adversity; many who came couldn’t afford to stay; others who came and stayed were with severe economic worries; others came and studied intermittently, gaining economic resources as the opportunity presented itself. Moore had been invited to author a colloquium publication for the American Mathematical Society and it appeared in 1932, entitled “Foun- dations of Point Set Theory.” On April 29, 1931 he was informed that he had been elected to membership in the National Academy of Sciences.7

6Bulletin, University of Texas, Austin, Texas, 1522, p. 6, No. 2242. 7Indeed, that evening a man named Patterson of the Zoology Department came by Moore’s house to inform him that Dr. Herman Joseph Muller of their department had 94 CHAPTER 4. RETURN TO TEXAS

Election to membership in the National Academy of Sciences came for Robert Lee Moore after a decade of most unusual productivity. In that single decade from 1920 to 1930, Moore published thirty papers. In 1932 he followed with the publication of his American Mathematical Society Collo- quium publication Foundations of Point Set Theory, in which he exposed results not before in print. Most of his papers during the decade following 1920 were of a fundamental character providing a basis for a school of mathematics in point set topology. Shortly before 1920, a paper jointly authored by R. L. Moore and J. R. Kline appeared in the Annals of Mathematics Vol. 20 (l919) 218-223 entitled, “On the most general plane closed point-set through which it is possible to pass a simple continuous arc.” Following is the introductory paragraph of that paper, along with the main theorem established. A set of points is said to be totally disconnected if it contains no connected subset consisting of more than one point. In 1905 L. Zoretti showed that every closed, bounded and totally disconnected set of points is a subset of a Cantorean line. In 1906 F. Riesz attempted to show that every such set of points is a subset of a simple continuous arc. Shortly thereafter Zoretti pointed out that Riesz’s argument was fallacious. He, however, left unsettled the question whether Riesz’s theorem was true or false. In 1910, in an article that contains no reference either to Riesz or to Zoretti, Denjoy indicated that this theorem could be proved with the use of certain ideas contained in a former paper of his own. We have not, however, succeeded in determining from his meager indications just what sort of argument he had in mind. At any rate, in order that a closed and bounded point-set should be a subset of a simple continuous arc it is of course not necessary that it should be totally disconnected. In the present paper we will establish the following result.

Theorem 4 In order that a closed and bounded pointset M should be a subset of a simple continuous arc it is necessary and suﬃcient that every closed, connected subset of M should be either a single point or a simple continuous arc t such that no point of t, with the except on of its endpoints is a limit point of M t. − Moore was attacking and settling diﬃcult questions. Recognition of Moore’s mathematical strength and taste came in many ways. In 1920 R. been elected to membership in the National Academy, only to have Moore declare that he too had been so elected. 95

L. Moore reviewed the second volume of Veblen and Young’s projective geometry. Even a non-mathematician might read that review and determine that serious questions of a fundamental sort were raised by the reviewer, while simultaneously stating, “The reviewer feels that the volume under review is a valuable addition to the as yet rather restricted list of advanced mathematical treatises of high grade published in America.”8 In his paper “Concerning simple continuous curves,” Moore considered removing the condition of boundedness from the extant deﬁnitions of simple continuous arcs. He took as his deﬁnition:

If A and B are two distinct points, a simple continuous arc from A to B is a closed, connected set of points M containing A and B such that

1. M A and M - B are connected, − 2. if P is any point of M distinct from A and from B

then M P is the sum of two mutually exclusive connected point-sets− neither of which contains a limit point of the other one.9

Interestingly he defined a “sect” of an arc, and pointed out that Halsted had used “sect” with different meaning. His second definition of a simple continuous arc, posed in that paper, fails to avoid a boundedness condition:

If A and B are two distinct points a simple continuous arc from A to B is a closed, connected, and bounded point-set containing A and B which is disconnected by the omission of an one of its points which is distinct from A and from B.

In another paper, “Concerning certain equicontinuous systems of curves,” Moore examined the meaning of a system of open curves being “equivalent to a system of parallel lines, stating

In order that a system G of open curves lying in a given plane S should be equivalent, from the standpoint of analysis situs, to a complete system of parallel lines in S it is not suﬃcient that through each point of S there should pass one and only one curve of the system G.10

8Bull. Amer. Math. Soc. 26 (1919-1920) 425. 9Concerning simple continuous curves. Trans. Amer. Math. Soc. 21 (1920) 334-335. 10Concerning certain equicontinuous systems of curves. Trans. Amer. Math. Soc. 22 (1921) 41. 96 CHAPTER 4. RETURN TO TEXAS

His observation led him to the following deﬁnitions concerning equicontinuity:

Deﬁnition 1 A system of curves G is equicontinuous with respect to a given point-set M if for every positive number there exists a positive number M such that if P and P are two points of M at a distance apart less than M and lying on a curve g of the system G then that arc of g which has P and P as endpoints lies wholly within some circle of radius .

Deﬁnition 2 A system of curves G is inversely equicontinuous with respect to a point-set M if for every positive number there exists a positive number δM such that if P1 and P2 are two points of M at a distance apart less than δM and lying on a curve g of the system G then that interval of g which has P1 11 and P2 as endpoints lies wholly within a circle of radius δM.

By 1922 Moore had appear in print his second paper in the Proceedings of the National Academy of Sciences. It was communicated by E. H. Moore and dealt with continuous curves in three dimensions. In part, Moore stated:

Schoenflies has shown that, in order that a closed, bounded and connected point-set, lying in a plane S, should be a continuous curve, it is necessary and sufficient that (1) if R is a domain complementary to M, every point of the boundary of R should be “accessible from all sides” with respect to R, and (2) if is any preassigned positive number, there do not exist infinitely many distinct domains, complementary to M, and all of diameter greater than . In the present paper I will exhibit examples of continuous curves, in space of three dimensions, which satisfy neither of these conditions.12

In a paper published that same year, in Fundamenta Mathematicae, Moore investigated the property of connectedness im kleinen, using the

11Concerning certain equicontinuous systems of curves.Trans. Amer. Math. Soc. 22 (1921) 42. 12On the relation of a continuous curve to its complementary domains in space of three dimensions. Proc. Nat. Acad. Sci. Vol. 8, No. 3, March 15, 1922, p. 33. 97 following deﬁnition (which diﬀered from Hahn’s requirement that x and p must be together in a closed and connected subset of M of diameter less than ):

A point-set M is said to be connected im kleinen at the point P of M if for every positive number there exists a positive number δP such that if X is a point of M at a distance from P less than δP then X and P lie together in a connected subset of M of diameter less than .13

At least, in 1922, Moore was considering such properties in a metric space and was not yet stating in print such properties in a more abstract setting. Continuing to have interest in analysis, in 1923, there appeared in the Bulletin of the American Mathematical Society a short paper which began with:

On page 92 of the 1907 edition of Hobson’s The Theory of Func- tions of a Real Variable, and again on page 113 of the second edition of the same treatise, there occurs the following statement14 (this statement will be called Proposition A):

Proposition A A non-dense closed set is enumerable if its complementary intervals are such that every one of them abuts on another one at each of its ends.

The two-page paper ended with Moore citing an example which he de- ﬁned and stating, “The existence of this example disproves Proposition A.” Testimony to Moore’s interest in analysis, and the mathematics supporting it, is given by the paper mentioned immediately above, in which he disproves statements in Hobson’s The Theory of Functions of a Real Vari- able. Further testimony is given by one of his students of the early 1920’s at Texas as she described Moore’s dealing with a student who had been enrolled in a course dealing with the “Foundations of Mathematics.”

I remember there was one poor boy in the class - I think he was a graduate student - he came from Southwestern and he thought he knew quite a bit about diﬀerent things in math.

13Concerning connectedness im kleinen and a related property. Fund. Math. Vol. 2, No. 4 (1921-23) p. 233. 14An uncountable, closed, and non-dense point set each of whose complementary intervals abuts on another one at each of its ends. Bull. Amer. Math. Soc. 29 (January 1923 ) 49. 98 CHAPTER 4. RETURN TO TEXAS

And Dr. Moore started pinning him down and he kept the boy so confused, he dropped the course. One day he said - Dr. Moore kept asking him things and he said, “Well, that’s not my ﬁeld.” And Dr. Moore said, “What is your ﬁeld?” And he said, “Analysis,” and Dr. Moore said, “Analysis, analysis’ What do you think this is?” I think that was a course in the density of sets.15

Moore continued his investigation of continuous curves, publishing several papers which dealt with properties of continuous curves in the plane and in three dimensions. His mathematical language was precise, brief, and lucid. Seldom did he suggest, with word pictures, a geometric representation of the mathematics involved. However, some of his doctoral students of later years were to become rather colorful in the terminology used to describe certain sets. A forerunner of that style of mathematical writing appears in a 1923 paper of Moore’s in which he states

Such a set of curves may form a surface containing a portion resembling a part of a coat with a pocket which, in addition to being attached as usual, is also sewed to the coat along its two “lateral edges.”16

The setting in which Moore taught and researched in the early 1920’s was a rather small university. The University of Texas in 1920-21 had a total enrollment of 6,888 students. That enrollment increased at a rate of almost 1000 students per year for the next few years, with 1923-24 enrollment 9250 and 1924-25 enrollment 10,461. In a setting, oﬀering students of that number, few sections of each course would be taught each year and the more advanced courses would frequently e taught on an every other-year basis. Not only was the population, through which Moore could search for talented students, a modest one, but the size of the institution allowed not for the development of a sequence of courses through which a student might expect to move for four years under a single instructor. By 1923 Robert Lee Moore had been promoted to Professor rank in the Department of Pure Mathematics. Also, by that time there was a Department of Applied Mathematics. Years before, while Moore was still an undergraduate student at the University of Texas, there had been a single School of Mathematics. Tradition was such that each school was under the direction of one Professor and no other person in that school would hold

15Personal conversation with Mrs. Blanche Bennett Grover, Houston, Texas, Decem- ber 18, 1970. 16On the generation of a simple surface by means of a set of equicontinuous curves. Fund. Math. Vol. 2, No. 4 (1921-23) p. 106. 99 that rank. While Halsted was at Texas, he held the post of Professor and, upon his departure, Porter was hired to fill that position. Benedict, another protege of Halsted was on the faculty at the time of Halsted’s departure and Porter’s return. Two future presidents of the University of Texas, Benedict and Calhoun, were on the mathematics faculty at the turn of the century. Partly because of the tradition that no school, or departments as they were later called, would offer more than one professorship, partly because Benedict’s training was in mathematics and astronomy, and partly because there was a desire on the part of some to offer Benedict a professorship, there would be formed in the early l900’s a separate Department of Applied Mathematics. Thus, by 1924-25, the Department of Applied Mathemat- ics offered courses which were “mainly designed for engineering students, but are open to academic students and count toward academic degrees.”17 Seven courses were offered, through a junior level advanced calculus course. By 1924 the tradition that a single department have only one professor had given way and the faculty of the Department of Applied Mathematics consisted of Professors Benedict and Calhoun; Associate Professor Rice; Adjunct Professors Michie, Cooper, Rupp; and Instructor Cleveland. That same year, 1924-25, the Department of Pure Mathematics faculty consisted of Professors Porter, Dodd, and Moore; Associate Professors Benedict and Ettlinger; Adjunct Professors Decherd, Batchelder, and Van- diver; and Instructors Horton, Holmes, Lubben, Stafford, Mullings, and Roberson. The course offerings totalled thirty. Moore was scheduled to teach “Introduction to Foundations of Geometry,” “Foundations of Math- ematics,” “Point Sets and Continuous Transformations,” and “Theory of Functions of Real Variables.” A student wishing to satisfy a Bachelor’s degree requirement in the College of Arts and Sciences in 1924 did so by completing a full year’s study in the Department of Pure Mathematics. One year later, 1925-1926, Moore had enlarged his teaching assignment to include calculus and “Research in Point-Set Theory.” By 19261927, the degree requirements had altered some, stating now that a student must offer six hours in mathematics. Moreover, the course listings in the Depart- ment of Applied Mathematics and the Department of Pure Mathematics no longer were designated as “applied mathematics” or “pure mathematics.” Less formal distinction was being made between the two departments. By the academic year 1927-1928, Moore had begun to formalize his sequence of courses. He now had listed in the catalog a calculus course, over his name, described thusly:

Recommended to students intending to continue mathematics

17University of Texas Bulletin, College of Arts and Sciences Catalog, University of Texas, Austin, Texas, 1924-25, p. 112. 100 CHAPTER 4. RETURN TO TEXAS

and to students of physics and chemistry. Prerequisite: Pure Mathematics 302. If before taking Pure Mathematics 13 the student has credit for nine or twelve semester hours in mathematics, Pure Mathematics 13 will count as three or six advanced semester hours.18

He continued to teach “Foundations of Mathematics,” “Point-Sets and Continuous Transformations?” and “Research in Point-Set Theory.” By the end of the decade of the 1920’s, Moore had not only accomplished much mathematics which was new and fundamental, he had also produced sound doctoral students and had begun developing a sequence of courses which allowed a student to progress under his direction throughout his college and university career. Of the fourteen faculty in the Pure Mathematics Department in 1929-1930, four had earned or would earn their doctoral degrees under Moore and one other doctoral student was with adjunct professor rank in the Department of Applied Mathematics. Moore was extremely serious in his dedication to mathematics and leg- ends would develop about his disregard of and independence from talented students who, in his opinion, were wasting their talents by not being serious enough about their mathematics. Some students, after having courses with Moore, would still decide to go elsewhere to continue their studies. Only in certain cases did he willingly observe students forsake him to study elsewhere. If he felt circumstances were such that the student could not perform well in Austin, he would not fight a student’s decision to go elsewhere. For instance, if he felt a student’s social habits were so demanding of his time that the student could not prosper, Moore might assist that student to continue his studies elsewhere. Or else, if Moore were convinced that a student had demanding and unnecessary interruptions on his time by family members who lives nearby, he might “send the student off” to study elsewhere. Indeed, if a student simply came to an irrevocable decision to go elsewhere, Moore would argue against it, if he thought the student were with promise, but then assist the student in locating elsewhere, once Moore became convinced that the student was committed to leaving. Moore surely was not above attempting to influence a student to work with him but, if that failed, or if it seemed that the student would prosper better elsewhere, he would offer assistance in gaining another post for the promising student. Of course, others from away who graduated from another of Moore’s students, would make their way to be about Moore for a period of time, by way of such means as a post doctorate.

18University of Texas Bulletin, College of Arts and Sciences Catalog, University of Texas, Austin, Texas, 1927-28 p. 187. 101

An example of one who left Texas and Moore in the 1920’s is offered by W. L. Ayres. Once Moore was determined in his own mind that Ayres was bent on continuing his study elsewhere, he assisted Ayres in gaining a post at the University of Pennsylvania. Ayres became Kline’s third doctoral student and, early in the experience of an unusually illustrious career as an academician, Ayres returned to Texas for a post doctoral assignment with Moore. Others, in addition to Ayres, of Kline’s doctoral students who spent a post doctoral year at Texas with Moore were his first student, H. M. Gehman, his fifth student Leo Zippin, and his sixth student, N. E. Rutt. Those who were at Texas in post doctoral positions were not at the center of Moore’s interest. His teaching efforts were toward his own students, not those who had already graduated under someone else. Indeed, even one of his own doctoral students, who established himself well as a mathematician, lamented after a post doctoral year at Texas that, “not once while I was there did Moore ask me to talk before his class.”19 Moore had his own manner of introducing subject matter to his students. He simply did not choose to allow another person that privilege with students working toward degrees under his direction. Though Moore was not inclined to invite others to lecture before his classes, he was known to counsel his students toward other faculty. There were a mixture of reasons for his counseling student toward another’s class. As one student put it, “Moore was jealous of a student’s time and would counsel him toward some faculty member whose course would not demand much of the student’s time.”20 At one point, E. W. Chittenden, a mathematician of considerable reputation visited Texas, teaching for a session. Moore was overheard in the auditorium, before his office, encouraging Chit- tenden to try his method of teaching. Chittenden protested, stating that he had certain material which he felt obliged to present. Moore continued, stating that perhaps Chittenden could order those theorems which he felt he should cover, from easiest to most difficult, making them sequential so that some would follow from others earlier in the sequence, and then present them to the class to see what transpired. He said, “Try it, just try it!”21 Chittenden did try it, and found it quite successful. As one was to later put it, “Chittenden raised a whole bunch of questions and the next time class met, the students began presenting proofs of Chittenden’s the-

19Personal conversation with R. L. Wilder, Santa Barbara, California, December 4, 1971. 20Personal conversation with Mrs. Lucille Whyburn, Charlottesville, Virginia, March 28, 1971. 21Personal conversation with Mrs. Lucille Whyburn, Charlottesville, Virginia, March 28, 1971. 102 CHAPTER 4. RETURN TO TEXAS orems and Chittenden never again regained the board.”22 Suspicion has long been held that Moore had “stacked” that class with the best students around and well knew their capability at proceeding on that basis “tried” by Chittenden. Moore had completed a full ten years at the University of Texas by 1930. It had been full in more than only a technical sense. He had begun the decade as a recognized mathematician who was beginning to produce strong students. By the end of the decade, he was an academician in full strength, a developer of much new mathematics, a recognized source of capable doctoral students, an exceptionally good teacher, and poised on the threshold of membership in the National Academy of Sciences. At almost that same time, as he approached full eﬀectiveness as an academic mathematician, that grave economic depression slammed across the nation, depleting the student population to which Moore would have access for the next few years. So ended the 1920’s and so began the 1930’s.

22Personal conversation with Ben Fitzpatrick, Jr., Auburn, Alabama, February 15, 1971. Chapter 5

The Mature Years: 1930-1953

The twenty-three years which began with 1930 saw many honors bestowed on Robert Lee Moore. Simultaneously, many changes were wrought within the mathematics program during those years, at the University of Texas. Prior to 1930, Moore had seen graduate under his direction, seven doctoral students, four of whom were granted their degree by the University of Texas. During the following twenty- three years, some nineteen additional students would earn their doctorate from the University of Texas under Moore’s direction. However, by the end of that period, much influence would have been gained by the Department of Applied Mathematics and Astronomy, with corresponding loss by the Department of Pure Mathematics, resulting finally in a joining of the two departments in 1953. The opening of the decade of the 1930’s found the nation still in the throes of a great national depression. Herbert Hoover was still in the White House. Some parts of the nation were still to feel the brunt of the agony of the nation. As a result of the depression years, students found themselves in college, often with inadequate means and sometimes because there was hardly any other activity available to them. Though the rate of growth of the university during the early 1930’s was not radically different from the 1920’s as the chart below illustrates, the certainty of a student that he would finish his line of study was often dependent upon whether any job opened to him. Finding a job, often the student would forego further academic activities to seek potential financial security.

1929-1930 5 774 1930-1931 6 041

103 104 CHAPTER 5. THE MATURE YEARS: 1930-1953

1931-1932 6 421 1932-1933 6 739 1933-1934 6 652 1934-1935 7 662 1935-1936 8 374 1936-1937 9 206 1937-1938 10 117 1938-1939 10 923 1939-1940 11 078

Though the student enrollment was increasing at about three hundred per year for the first three years of the 1930’s there were some one hundred fewer enrolled in 1934-1935 than in 1933-1934. For the academic year 1930-1931 there were nine faculty in the Applied Mathematics Department, offering some twenty courses. At the same time, some fourteen Pure Math- ematics faculty offered forty-one courses. By 1934-1935 some eight Applied Mathematics faculty taught twenty-four courses and fourteen (though one was on leave) Pure Mathematics faculty taught forty-two courses. How- ever, by 1936-1937 ten applied faculty were teaching twenty-nine courses and fifteen pure mathematics faculty offered thirty-seven courses. With the beginning of the next decade, 1940-1941, fourteen applied mathematicians taught thirty-five courses while eighteen pure mathematics faculty taught thirty-seven courses. Toward the end of the decade of the 1940’s applied mathematics faculty would number twenty- seven, along with twenty-five teaching fellows while the pure mathematics faculty numbered nine with twelve instructors. The applied mathematics offerings clearly were finding audience among the students, to the point that the Applied Mathematics Department grew to match, and then surpass, the size of the Pure Mathe- matics Department in terms of faculty number and student enrollment. In 1930-1931 the Applied Mathematics Department included Benedict and Calhoun among its eight faculty. Also among that faculty was one of Moore’s doctoral products, Cleveland. Benedict was then President of the university and would serve until his death in 1937. Calhoun would succeed him in office and serve for two years. The department was not without influence. Of the fourteen pure mathematics faculty in 1930-1931 two (Lubben and Roberts) were Moore doctoral students still on the faculty. Among the four instructors were Hamilton, Klipple, and Vickery. The latter two were to be Moore products and, though Hamilton did not take his degree with Moore, Moore’s influence on him is clear from his long career as a point set topologist. That year, Moore offered Advanced Calculus, Introduction 105 to the Foundations of Geometry, Foundations of Mathematics, Point Sets and Continuous Transformations, and Research in Point Set Topology. Moore was involved during these years with a monumental effort of writing Volume XIII of The American Mathematical Society Colloquium Publications, Foundations of Point Set Theory. Included in the preface of that book is the sentence:

With the intention of reserving them for publication in full, for the ﬁrst time, in the present volume, the author has, during the last four or ﬁve years, allowed to accumulate, certain results which normally he would have published from time to time in separate articles.

Even so, Moore’s rate of publication remained strong. He had published two papers in 1929 and, following publication of his colloquium volume, published at the rate of almost one paper a year until 1946. It was the period following World War II that so many students found their way to college campuses. Indeed, the enrollment jumped at the University of Texas from 9400 in 1944-1945 to more than 15,000 in 1945-1946. In the presence of that tremendous enrollment increase, the permanent members of the faculty of the Pure Mathematics Department remained almost without change from that faculty of the pre-war years. In 1940-1941 that faculty had included nine at the instructor rank; that is, graduate students who were seeking degrees. The post-war years saw the number of such instructor- ships increase to no more than thirteen. Thus, even though the university had almost doubled in size from 1944-1945 to 1945-1946, the faculty of the Pure Mathematics Department experienced no similar increase. Though it is true that much of the increase in enrollment found its way to the applied offerings, there did occur a sizeable jump in the enrollment of the Pure Mathematics offerings. That classroom demand, along with a greater number of graduate students seeking doctorates, radically increased the demands on Moore and others in the faculty at Texas. In addition to his research, teaching, direction of students and other university involvements, Moore received honors which carried additional responsibilities. In 1931, 1932, 1933, 1934, and 1944 he served on committees of the American Mathematical Society to nominate officers and members of the Council. He served as President of the American Mathematical Society in 1937-1938 and was in 1947 Vice President of the American Asso- ciation for the Advancement of Science, a post held earlier by George Bruce Halsted. Though his years of the 1930’s were busy and active, with many accomplishments and duties, it was the post World War II years which placed 106 CHAPTER 5. THE MATURE YEARS: 1930-1953 extreme demands on his time and ingenuity. There was an abundance of students, many serious to be about their interrupted lives, others “putting in their time like good G.I’s,”1 but all presenting Moore with raw talent that offered potential development into mathematicians. The returning veterans made up a major part of the increase in enrollments felt by universities across the nation. Some selected the University of Texas for reasons unrelated to academic strength. For instance, one veteran recalled that

I had been stationed at Randolph Field, just outside of San Antonio, and had become much enamored of this part of the country. And a portion of my inclination toward, then, coming to the University of Texas when I got out of the service was dictated by fond acquaintances with people that I had learned to know socially in this part of the country, coupled with what eventually boiled down to an academic choice between the Uni- versity of Texas and Ohio State. And I think it was the climate which may have turned the tide in favor of Texas. The climate in the sense that, being realistic about my ﬁnancial status, it would be much less expensive to buy clothing in this part of the country than any place in the north, where one has to be equipped with summer and, also, heavy winter clothes.2

That same veteran recalled his initial experience with Moore:

I was being exposed to Dr. Moore for the first time in his senior level course. And in there the atmosphere was a lot different because it was the notion of his announcing what we should use as an axiom and then his proposing certain things as theorems for us to try to prove. And he posed the first one and I shall probably remember that theorem to my dying day. The second class meeting with Moore, he asked if anyone had a proof of that theorem, and I volunteered that I did. He had me go to the blackboard and I used up the whole class period and I had great logical holes in my argument and my proof just was not a proof; it wouldn’t work. And the period was over and class recessed with no comment from Moore, except that, “It looks as if that isn’t quite an argument.” The following class meeting, the third of the semester, “Does anyone have that theorem?” Again, I volunteered. And again, it was a long

1Personal conversation with John S. Mac Nerney, Houston, Texas, June 14, 1971. 2Personal conversation with John S. Mac Nerney, Houston, Texas, June 14, 1971. 107

and arduous class meeting with me at the board, attempting to prove this theorem and still having difficulties with it; seeming to me, to have difficulties in communicating my ideas to Moore. The upshot of it was that this was another class period used up with no proof of that theorem. Well, this I found very, very challenging and, immediately after that class meeting, I went home and took out some fresh paper and sat down and sketched out very, very carefully an argument for this theorem and checked it out in every detail and wrote it out word for word, verbatim, my proof of the theorem. This time I was sure that I’d had it. When we met next class meeting, in fact, I did present it, it was an argument, it was a proof of the theorem. Dr. Moore was convinced, then, I thought, that the theorem was true, which, of course, he had known all along. And when I got all through, using, I think, about thirty-five or forty minutes of that class period, at the blackboard and asking whether or not there were any questions, Dr. Moore merely smiled slightly and said, “Well, that’s a good argument.” And I took my seat and he proceeded to ask other questions of the class. And that was fairly well the pattern of that first year of my experience with Moore. But, from then on out, I made it a point, and I believe that Dr. Moore realized that I was making it a point, that when I announced that I had a proof of a theorem, he could be pretty sure that, in fact, I did. And so those first few class meetings in Dr. Moore’s course, though at the time they might have seemed to an observer like lots of time wasted, this was the valuable time spent in letting a student make mistakes and learning to correct his own mistakes.3

Even in the presence of almost overwhelming numbers of students, the permanent faculty of the Pure Mathematics Department did not relegate the teaching of lower level courses exclusively to graduate students, as has been the custom in many “major” institutions. Indeed, a graduate student recalls that, in 1948,

. . . it was rather infrequent that there would be a section of Analytic Geometry which would not be taught by some member of the permanent staﬀ in the department. There was the tradition through those years that analytic geometry, in part, but calculus, in general, was regarded as really the most diﬃ- cult course to teach in the undergraduate curriculum and those

3Personal conversation with John S. Mac Nerney, Houston, Texas, June 14, 1971. 108 CHAPTER 5. THE MATURE YEARS: 1930-1953

of us who were teaching as graduate students would mainly get to teach undergraduates prior to the calculus, but then, occasionally, to teach courses in advanced calculus and diﬀerential equations courses which would normally follow the ﬁrst year calculus sequence4

The students who graduated under Moore’s direction during the 1930’s were: C. M. Cleveland, J. L. Dorroh, C. . Vickery, E. C. Klipple, R. E. Basye, and F. B. Jones. Interestingly enough, they came to Moore from hometowns of Union, Mississippi; Rosebud, Texas; Quitman, Texas; Cuero, Texas; Kansas City, Missouri; and Cisco, Texas. Just as debates occur as to the “greatest” boxer of all time, or the greatest baseball team of all time, so have debates sprung up over from which period Moore graduated the best students. On the floor of a general faculty meeting, at the University of Texas in 1954, it was claimed that his 1945-1953 group was the very best to have graduated under his direction at that time.5 Those who graduated under his direction between 1940 and 1945 were R. L. Swain, R. H. Sorgenfrey, H. C. Miller, and G. S. Young. Graduating after then and before 1954 were R. H. Bing, E. E. Moise, R. D. Anderson, M. E. Rudin, C. E. Burgess, B. J. Ball, Eldon Dyer, M. E. Hamstrom, and J. M. Slye. Moore’s teaching talents again evidenced themselves as at least one of the 1945-1953 group was to gain membership in the National Academy of Sciences (in 1965). He was R. H. Bing, who returned to Texas as a high school teacher and found Moore’s classes particularly exciting. Moore took his students as he found them, recognizing talent where it was, and offering it opportunity to flourish in a setting which demanded independent mental activity of high order. The surplus of students of the post-war years allowed Moore the opportunity to regularly offer formal courses at each level, generating a clientele for the offering sequel to each. Competition between students was keen and, on occasion, a student’s dissertation effort, when presented in class, demoralized temporarily others who were yet to gain a result sufficiently strong. Although the two decades following 1930 were filled with much research and teaching activity for Moore, his voice began being heard more forcefully within the chambers of the General Faculty meetings. In 1937 a discussion on the floor of the general faculty occurred which dealt with the role of a department head and Budget Council of a department. Moore objected to a committee report that was before the house in the form of a motion

4Personal conversation with John S. Mac Nerney, Houston, Texas, June 14, 1971. 5Minutes of the General Faculty, University of Texas, Austin, Texas, March 9, 1954, p. 6288. 109 oﬀering, within his statement of objection, a statement concerning his own philosophy as regards promotion.

He (Moore) read the following extract from the Committee’s report: In large departments the council is numerous, and from the standpoint of efficient, integrated control the consequent diffusion of responsibility cannot be justi- fied for it inevitably leads to ill-defined and vacillating policies and often to an evasion of responsibility. He declared that if this meant anything it meant in simple language that the departments were lacking in efficiency because of their organization, and declared that he doubted very much if a department would become more efficient through a change in its organization and maintained that the University was suffering more from the lack of ideals than from the lack of efficiency. He was of the opinion that the standing of a university was apt to be more adversely affected by too rapid promotions than by too slow promotions; and that the new scheme would undoubtedly accelerate promotions of persons who were not deserving, since it was easier to get the consent of one man than that of five or six. “A man with sole responsibility may be placed in a position where he must give way even against his own convictions. I have heard it said that this scheme would be a fine thing for the women, because no one man could stand up against their appeal. I believe that the evils constantly referred to by the Committee are grossly exaggerated and in many departments they don’t exist at all.”6

Following defeat of the motion in question, a few days later an amendment oﬀered to another motion met with this response from Moore:

Mr. R. L. Moore said that he would vote against the Law amendment not only on the ground that it required the chairman to be a member of the Budget Council, but also because it would merely confuse the issue. He hoped that the Faculty would turn its thumbs down on the whole recommendation of the Committee. “Why not leave the chairmanship as it is? Why should we constantly change from one thing to another? Why

6Minutes of the General Faculty, University of Texas, Austin, Texas, March 22, 1937, p. 1207. 110 CHAPTER 5. THE MATURE YEARS: 1930-1953

should the chairman be a full professor? The Committee amendment provides for virtually the same thing that the Faculty decisively rejected a few days ago. It seems to me that when they discover they can’t get what they want in one way, they try to get it in another – if not in one form, in another form.’7

Moore continued to hold fast to positions he felt were correct. It may well have been the case that, as a committee member, he was not one with which it would be easy to gain a consensus of opinion. He was known to withhold his signature from committee reports which he felt did “not put the case strongly enough.”8 Student lore had it that, at one point, becoming impatient with another committee member’s nonserious behavior, he threatened a spanking if the behavior didn’t cease, and then fulﬁlled that threat when it didn’t. Discussions were held in the late 1930’s regarding the making of appointments at the University of Texas. Proposed was a motion which in part stated:

Departments should make every effort to secure persons from different universities, different sections of the United States, and from other countries. It is desirable to set up the policy of advancing no one to a professorial rank who has not studied outside the State.9

Moore is recorded as having diﬀered with that restriction on appointments:

Some departments, he believed, would be better off if they had more inbreeding. He illustrated his point from the University of Chicago declaring that that institution had been the loser in a good many instances in which Chicago trained men had been permitted to go to other institutions and men trained by other institutions had been appointed to the vacancies. He thought it was absurd to adopt a hard and fixed policy on appointments and inquired why any first-class man should be excluded no matter where he had received his training.10

7Minutes of the General Faculty, University of Texas, Austin, Texas, April 6, 1937, p. 1223. 8Minutes of the General Faculty, University of Texas, Austin, Texas, March 9, 1954, p. 6288. 9Minutes of the General Faculty, University of Texas, Austin, Texas, November 9, 1937, p. 1320. 10Minutes of the General Faculty, University of Texas, Austin, Texas, November 9, 1937, p. 1321. 111

Soon thereafter, discussions were held on the ﬂoor of the general faculty designed to set standards by which the qualiﬁcation and service of teachers might be judged. In response to a committee report which favored such standards and designated some of them, Moore

. . . pointed out that if the amendment were adopted it would be necessary for the chairman to investigate annually the personality and character of each member of the staff. He said that he would vote for it, but only for the purpose of making the report of the Committee as ridiculous as possible and increasing the prospects of its defeat. In his opinion the Faculty ought to vote down decisively the entire section. Moore read a list of terms used by the proponents of the report which he had taken down during the debate: “rating scale,” “efficiency records,” “government employees,” and “personnel management.” He thought that these were foreign to a real university. There was a time when a faculty member resented being called an employee; he hoped that that time was still with us. He criticized the suggestion that a chairman be required to send in a report on every member of the budget council and declared that the committee was proposing indirectly something which the General Faculty had already voted down by an overwhelming vote, referring to the action of the General Faculty last year when it rejected the recommendation to establish powerful departmental heads. He said that if the Faculty adopted this recommendation every member of the department would know that the chairman must annually report upon him and that unless the member was a man of independence, he would immediately proceed to cultivate the chairman for self advancement. “What is going to be put into the records–into the permanent records–which is to be kept by the vice-president? Is the record going to say that this year a man’s teaching is good and last year it was bad? There are some people on this faculty who say that they are in sympathy with the general objectives of the report but op- pose certain details. Yet it is hard to find out from them which recommendations of the report they actually approve.” 11

Moore continued to speak out on questions which he felt would retard the university’s eﬀorts toward quality. A discussion over tenure regulations occurred on the ﬂoor of the General Faculty, with Moore stating:

11Minutes of the General Faculty, University of Texas, Austin, Texas, November 9, 1937, pp. 1323-1324. 112 CHAPTER 5. THE MATURE YEARS: 1930-1953

The first three sentences of the third paragraph of Section 3 read as follows: “The term of appointment of an instructor will be one year. In the event of decision not to reappoint an instructor for a second year, at least three months notice will be given him. Thereafter one year’s notice will be given him.” The sixth and seventh sentences read “The term of appointment of an assistant professor will be two years. In the event of decision not be reappoint an assistant professor, one year’s notice will be given him.” I wonder how many, if any, of you realize that this means that if an instructor is appointed initially for one long session then, before the end of that long -session, the budget council of his department must decide either (1) to inform him that it will not recommend his reappointment or (2) to inform him, in effect, that he will be given a terminal appointment of one additional year in which to secure another position or (3) to inform him that he will be reappointed for a second term, this time of two years? This comes pretty close to being a disguised scheme for having an instructor’s second term one of two years, regardless of any previous statement to the contrary. Similarly it seems clear that this committee recommends, in effect, that the first term of appointment of an assistant professor shall be one year (followed by a terminal appointment for one additional year in case of a decision not to reappoint him for a regular second term, that is to say for a second term which is not specified in advance to be terminal) and that a regular second term shall be for three years so that after an assistant professor has been here for one long session the budget council is obligated to decide whether

to notify him that he will be given a terminal appointment • of one more year in which to ﬁnd another position or to recommend that he be appointed for a regular second • term, in which latter case he will have tenure for four more years.

I emphasize that I mean tenure for four more years, not just for three. And there is considerable likelihood that, after the ﬁrst three of these four years, it will extend for the rest of his life or until the date of his retirement. I address myself now, not to those who are more concerned with having regulations adopted to prevent budget councils from “exploiting” the mentally underprivileged than they are with taking steps towards 113

making this a university of the first class. I address myself, not to them, but to those of you, if there are any, who have some conception of what a university of the first class really is and who are determined to try to ensure that, at least as far as your own departments are concerned, it will really become one, if it is not even now, and that, if it is now then it will remain that way if it is in your power to have it so. I ask you are you willing to have the Board of Regents requested to adopt regulations requiring that if an assistant professor, perhaps a man who has just received his Ph.D. degree, is added to your department, then, before you have known him for more than about nine months, you must decided definitely to drop him or decide definitely to keep him in your department for four more years? In my opinion, if this legislation is adopted, its adoption will be one more step towards making this university safe for mediocrity. And there is already entirely too much of that sort of security here.12

Shortly after discussions over tenure, the question of faculty promotion standards became a focal point of faculty debate. Proposed as new procedures for promotion were recommendations including:

It shall be the duty of budget councils of various teaching departments to recommend or not recommend the promotion of instructors and assistant professors according to the following procedure. Before a member of the teaching staff shall have completed four years of service as an instructor (full time) the Budget Council of his department must consider the matter of promoting him to the rank next above. In case the Budget Council is unwilling to recommend such promotion, the procedure shall be as follows: The staff member will be notified by the Chairman of the Budget Council that he is not likely to receive further promotion and will be given a terminal appointment for one year in which to secure another connection.13

Rationale presented in support of the proposed plan included:

1. That the best of our instructors and assistant professors desire an orderly and fairly prompt recognition of their merits, and that such

12Minutes of the General Faculty, University of Texas, Austin, Texas, May 20, 1943, pp. 3163-3164. 13Minutes of the General Faculty, University of Texas, Austin, Texas, May 15, 1945, p. 3150. 114 CHAPTER 5. THE MATURE YEARS: 1930-1953

orderly and reasonably prompt promotion can not be executed without some system of elimination; and

2. That indeﬁnite tenure should be achieved on the basis of merit and a not too long apprenticeship in teaching service.14

Following a statement in support of the recommended procedure, Moore addressed the faculty.

I do not claim to know how to deﬁne a university of the ﬁrst class, but I wonder whether anyone here would challenge the assertion that no university is of that class unless

a very substantial amount of really fundamental research • of a high order is carried on by members of its faculty, and there are some members of its faculty who are intensely • on the alert to discover and develop outstanding research ability on the part of their students and who are both capable of recognizing such ability in the early stages of its manifestation and of developing it when it is discovered.

If no objection is made to this assertion then I wonder on what ground, if any, one could object to the assertion that if a faculty member carries on no fundamental research worthy of the name and neither discovers nor develops the ability of any of his students to do so, then he does not belong in a university of the first class unless perhaps in some sort of secondary sense. Certainly I think it will be admitted that a university can not possibly be of the first class if all of its faculty members are of this secondary type and it can not be if too many of them are of this type unless the remaining members are of such outstanding character as to make up by very extraordinary quality what they lack in numbers in which latter case, unless their influence, as well as the quality of their work, is out of proportion to their numbers, there is grave danger that the university will not maintain its position after they have passed from the scene. Now let us consider the probable effect of this automatic promotion scheme if it is put into operation here at the University of Texas where we already have on the faculty entirely too many people who do not belong in a university of the first class. I ask

14Minutes of the General Faculty, University of Texas, Austin, Texas, May 15, 1945, p. 3150. 115 you to picture to yourself a certain type of hypothetical instructor. His personal qualities ensure that he will get along well with students. He meets and dismisses his classes promptly, grades quiz and examination papers carefully and conscientiously and returns them promptly and makes out reports neatly and gets them in on time. If there is a grade curve he follows it. He attends all meetings of committees of which he is a member and usually, if not always, votes with the majority. His dean receives no complaints from his students or the parents of his students. He gives freely of his time to students who come to him for assistance or consultation. The chairman of his department receives frequent requests from students who wish to transfer into his classes and none from any who wish to transfer out of them. He is a good example of what is ordinarily thought of as a satisfactory teacher of students of average ability. He has a Ph.D. degree from a leading university where his dissertation was supervised by an able investigator who supplied him with enough partly worked out problems to furnish material for the publication of several papers beyond his thesis. But he does not have the ability to continue under his own power when this material is exhausted and after ﬁve or six years have gone by he will never produce anything that would be accepted for publication in any reputable journal of national standing though he may keep up some appearance of activity through the medium of newspapers, university bulletins, and other local or semilocal publications and perhaps the radio. I think that after such a person has been an instructor here for four years it would be a very unusual budget council that would refuse to recommend his promotion if the only alternative were to drop him one year later and I believe that, even of those who would otherwise be inclined to do so, a considerable percentage would be deterred by the thought that his being dropped after a trial period of four years would probably make it diﬃcult, if not impossible, for him to secure a suitable position elsewhere. And if he is promoted to an assistant professorship, does anyone think that four years later, after eight years of service in this university, he will be dropped instead of being promoted to an associate professorship? And who, if anyone, thinks that if such a man comes to the University of Texas as an instructor at the age of 26 and is made an associate professor at the age of 34, then it is very likely that he will remain at that rank for the remainder of his life? The chances are, I think, that if this rule is adopted, 116 CHAPTER 5. THE MATURE YEARS: 1930-1953

then by the time he is 42, f not before then, such an individual will be a full professor and as a member of a budget council will vote for the promotion of more people of his own sort and later they will vote for still others and so on. As to the assertion that it is not fair to retain a man as an instructor for more than • four years without promoting him, and that the various departments of the university should have uni- • form standards with regard to promotion, if a speaker makes such an assertion without specifying just what sort of uniform standards he has in mind and he has a controlling influence in the budget council of his own department, then I suggest that you consider that department carefully and ask yourself what sort of institution this would be if the standards of all of its departments were as low as those of that one. I do not believe that this university will ever be of the first class (except with reference to a few departments that somehow manage to maintain high standards in spite of the obstacles imposed by such people) if it is dominated by the ideals of those who are more concerned with uniformity of standards and “fair” treatment of the mediocre than they are with the establishment and maintenance of high standards and the discovery and fostering of outstanding ability. If there is anyone here who would say “I agree with all you say about budget councils. I agree that strong budget councils need no such rule as this, that, rule or no rule, they will not retain mediocre instructors except temporarily and then only under very unusual circumstances such as those under which we may be forced to operate for a period of unknown duration after the war, and I agree with that, in the great majority of cases, a mediocre budget council will recommend the promotion of a mediocre instructor at the end of four years if the only alternative is to drop him, but you are overlooking the fact that the administration does not have to approve the recommendation of such a budget council” - if a person makes that statement, I would ask how much experience he has had with administrators during the last 24 years. If he has had as much as I have had at this university and still insists that administrators usually pro- mote only those who measure up to high standards, I would ask him how then does he account for the existence on this faculty 117 of so many who have been promoted from the rank of instructor all the way up to that of full professor but who clearly do not belong in a university of the first class unless in the secondary sense previously indicated. If, on the other hand, he says that he agrees with me that most administrators would approve the recommendation of a budget council that a mediocre individual be promoted if the only alternative were to drop him, but there is now, in this university, an administrator who would not do so, then I would reply that if the university is lucky enough to have such an administrator now, for probably the very first time in its history, then not only does such an administrator need no such rule but • the adoption of such a rule would have the definitely harm- ful effect of preventing him from judging each individual case on its own merits, and the chances are many to one that his successor would rather • have a mechanical rule to relieve him of the responsibility of so exercising his judgment and under the administration of such a successor, if not under his own, this rule, if adopted will operate to ensure and speed up the promotion of mediocre instructors and assistant professors, lead to a degradation of the titles of assistant professor and associate professor and eventually to that of full professor with a consequent further deterioration of budget councils and a resulting still further lowering of standards with regard to promotion. Recently I had a conversation with a very influential member of this faculty. I referred to certain individuals who have served for unusually long periods as instructors or assistant professors and raised the question whether it is not true that if this proposed rule had been in operation when they were first appointed they would still be here, but as associate professors or full professors instead of as instructors or assistant professors. He indicated that that would be satisfactory to him! I understand that he is one of the sponsors of this automatic promotion scheme. I believe that if it is adopted the issue will be decided by the vote of those who agree with him whether or not they are frank enough to admit it. I think that if it is adopted, its adoption will furnish a striking example of the danger residing in the 118 CHAPTER 5. THE MATURE YEARS: 1930-1953

presence on a university faculty of too many people who do not belong in a university of the first class. Entirely too many of them are likely to support legislation calculated to favor their own interests at the expense of the welfare of the institution and to lead to the propagation of more of their own kind to its still further detriment. I am unqualifiedly against this mechanical scheme regardless of whether it is proposed to put it into effect now or a hundred years from now. But to think of anyone’s proposing it now of all time’s! It seems extremely likely that after the end of the war with Japan there will be increased enrollment in universities without a proportionately increased supply of men of the right sort to staff them. And it seems extremely likely that there will then be unusually keen-competition for those who are available, particularly in the sciences, including mathematics. What can a science department do under these conditions if it wishes to maintain its standards? If it can not immediately secure enough men of the sort that it would be willing to retain permanently then it might bring in a number of instructors and perhaps continue to employ some already here who would, under ordinary circumstances, have been dropped before now, hoping that some of them may measure up to its standards but intending, if necessary, to keep some of them only until the available supply of properly trained and properly equipped men comes so near to equalling the demand that these people can be replaced by men of a different calibre. And will anyone here undertake to guarantee that the supply will equal the demand in the next five or six years. Do not lose sight of the fact that about eight or more years often elapses before an eighteen year old student receives the degree of doctor of philosophy. 15

No one who heard Moore’s comments would ever charge him with attempting to curry favor among his opponents to gain passage of a motion he desired! Moore’s activities on the floor of the General Faculty meetings ranged from expressions of philosophy of higher education to debates over the salaries and teaching effectiveness of graduate students. Moore, and others of the Pure Mathematics Department, consistently attempted to produce effective academicians. As well as producing researchers, they strove

15Minutes of the General Faculty, University of Texas, Austin, Texas, May 15, 1945, p. 3150. 119 to produce teachers. Following World War II, the Department of Pure Mathematics had practiced the hiring of graduate students as full-time instructors, each teaching four courses. Such a person, pursuing his degree simultaneously, would ﬁnd his circumstances much as they would be upon graduation, full-time teaching with a further obligation to be research active. It was good, strong training. A proposal was made to restrict the teaching of graduate students to half-time (two courses), paying them accordingly, but recognizing three categories of wages, depending upon the degree they held. When the Faculty Council took up the question of reorganizing the role of graduate student teachers, Moore asked to speak to The Council. As the discussion was underway, Moore gained the ﬂoor to state:

. . . that some graduate students were better teachers than full-time instructors, and that if highly qualiﬁed, ought not to be discriminated against by an inferior title and a lower rate of pay. He criticized the inclusion of the M. A. degree requirement and declared that in his department graduate students were not encouraged to take the M. A. degree which in his opinion constituted in many instances a sheer waste of time. If a graduate student displayed real scholarship, he ought to go right on to the Ph.D. degree and if he did not give evidence of sound scholarship, he ought to be sent elsewhere. 16

Calhoun, who had followed Benedict in oﬃce as President for two years, and who was a member of the Department of Applied Mathematics, recalled having been a student under Moore, before Moore left Texas to continue his graduate studies at Chicago. Calhoun added that he

. . . agreed with the statement that Professor Moore had made earlier that a person need not necessarily have a degree or a high title to be an eﬀective teacher. He said that he had taken a course under Professor Moore himself before Professor Moore had received either the Ph.D. degree or the full professor’s title and it had been one of the best courses that he had ever had. He stated also that he did not see why any limit should be placed upon the amount of class work which a graduate student, teaching part-time, could take. He said that that should depend upon the graduate student’s own ability and desire. He thought, moreover, that much of the discussion was out of order in that it

16Minutes of the General Faculty, University of Texas, Austin, Texas, April 9, 1947, p. 4398. 120 CHAPTER 5. THE MATURE YEARS: 1930-1953

concerned itself with questions other than that of the abolition of the parttime instructorship title and the substitution of the teaching fellowship classiﬁcation.17

Increase in student enrollment, and a desire on the part of some to join the Departments of Pure Mathematics and Applied Mathematics, led to there being constructed a building to house the two departments. Though feared by the Pure Mathematics group as a move to eventually join the two departments by weight of physical proximity, it was agreed that such housing would occur. Indeed, in 1949 a committee, consisting of C. M. Cleveland, H. J. Ettlinger, and C. P. Boner, was formed to consider naming the new building. Shortly thereafter the committee reported:

C. P. Boner and C. M. Cleveland recommend that the name be Benedict Hall. Dr. H. J. Ettlinger, the third member of the Committee, abstains from voting but wishes to oﬀer no objection to the recommendation of the other members of the Com- mittee. C. P. Boner C. M. Cleveland, Chairman

Mr. Boner then moved that the report be adopted, and the motion was seconded. Mr. J. A. Fitzgerald moved to amend by substituting for the name Benedict Hall, “Harry Yandell Bene- dict Hall.” There was no second to the amendment and the original motion was put and carried.18

Moore’s educational philosophy is reasonably well apparent from the comments he offered on the floor of the faculty, as he discussed departmental governance, promotion standards, status of graduate students, and his utter demand for quality. One further evidence of his philosophy of education and teaching is offered by comments he made in attacking a proposed core curriculum for students, supposedly to guarantee the breadth of their training. He stated:

Now for some excerpts from the Report of the Committee on the Core Curriculum. From pages 14 and 15 of that Report: “Specialization, Departmentalization, and rigid separation of

17Minutes of the General Faculty, University of Texas, Austin, Texas, April 9, 1947, p. 4399. 18Minutes of the General Faculty, University of Texas, Austin, Texas, November 1949, p. 5200. 121 subject-matter areas have all too often destroyed balance and unity. Objectives of a core curriculum must therefore be considered on a basis broader than that provided by departmental subject matter or specialized educational aims.” From page 21: “The departmental pattern of course offerings, however appropriate for advanced study, has handicapped efforts to integrate knowledge in broad areas such as science, social science, and the American intellectual and cultural heritage.” From page 31: “Many of the courses are of great scope. It will not be enough to supplement the interests of one department with those of another. Interests will have to be integrated and extended.” From the same page: “A considerable number of faculty members will have to devote a major part of their time to teaching in the fields of common interest instead of in fields of special interest.” From page 32: “The core curriculum courses suggested in this report are, for the most part, not courses that can be offered sepa- rately by individual departments.” Members of the faculties of the professional schools please note the following quotation from the same page: “Problems of realignment of requirements for specialization will have to be considered for the several col- leges and schools of the university.” And now, the very next sentence: “It seems almost certain that the over-all development will take several years and that some part of the program will have to be instituted one year and other parts in succes- sive years.” Consider this sentence in connection with Professor Graham’s statement that the Committee had not recommended any composite courses that would cross departmental lines that it had preferred to “take one step at a time,” and the statement that Dean Haskew thought the advantages of his amendment were It would provide the incentive for devising composite of- • ferings which would cover two or three areas mentioned in the report and something else. • He had long felt that after the defeat of the Core Curriculum the Graham Committee was trying to ease it in by way of the General Faculty. He wondered what the skeptics who disagreed with him then were thinking now. He concluded by stating as follows: Let X represent the aims of those who are in favor 122 CHAPTER 5. THE MATURE YEARS: 1930-1953

of the adoption either of the Graham Report of the Haskew Substitute and let Y denote the aims of those who are in favor of the adoption of the Report of the Core Curriculum Committee. Then the X-Y classiﬁ- cation is an incomplete disjunction. For those of you who are not so liberally educated, let me explain that this means that the aims of the people in these two groups have something in common.19

At a later date when the General Faculty was again discussing this issue he addressed the faculty as follows:

According to the Minutes of the last meeting of this Faculty, Dr. Law indicated that he thought that I believed in a Core Curricu- lum, that every curriculum had a core and I believed in some curriculum and therefore I believed in a Core Curriculum. Now I would have expected Dr. Law to know very well that, when I indicated that I thought the Committee on Liberal Educa- tion Requirements was laying the foundations for another Core Curriculum, I meant for a curriculum of the same type as the one titled “The Core Curriculum” and described in a pamphlet circulated in 1952 under the title “Report of the Committee on the Core Curriculum.” It might be difficult, or impossible, for me to draw a sharp line of demarcation and say that every curriculum on one side of the line is of the Core Curriculum type and no one on the other side is of that type. But, for those, if there are any, who do not have even a rough idea of what I mean, perhaps I might give some suggestion of my meaning in the following manner. I ask you to picture to yourself two hypothetical professors, Professor A and Professor B. Professor A thinks that every field in his department is equipped with a grave in which every one who specializes in that field sooner or later lies down, sinking lower and lower as his specialization continues; and this professor warns his students not only that there is such a grave for every field in his department but that the same is true for all other departments and if a student wants to stay above ground as long as possible he had better tread lightly on the surface, making horizontal excursions into

19Minutes of the General Faculty, University of Texas, Austin, Texas, May 3, 1955, pp. 6530-6531. 123

all possible fields thus becoming a well integrated, well balanced, horizontally well rounded whole man, with a really liberal education which will enable him to carry on a conversation in any field with any person whose acquaintance with no field as a vertical component. I would certainly expect Professor A to be strongly in favor of a curriculum of the Core Curriculum type. Let us now turn to hypothetical Professor B. He knows that there still lie ahead great advances to be made in his field, advances to some of which he would like to contribute both through his own work and through that of his students. He feels that a whole lifetime would not be sufficient for all that he would like to do but he knows that if he does not succeed in accomplishing a considerable fraction of it, the failure will be due to his lack of ability and to nothing else and he will derive no comfort whatsoever from any thought that there are many fields concerning which he knows something but to whose advancement he has contributed nothing of any importance. I think there is little or no chance that Professor B would have any sympathy whatsoever with any curriculum of the Core Cur- riculum type.20

Moore left no doubt as to whether he was an “A” faculty member! In- deed, his preparation and delivery must have made him a feared opponent.

20Minutes of the General Faculty, University of Texas, Austin, Texas, May 17, 1955, p. 6575. 124 CHAPTER 5. THE MATURE YEARS: 1930-1953 Chapter 6

The Years of “Modiﬁed Service”

By 1953 Robert Lee Moore had been teaching for over half a century. His teaching method was mature and the setting in which he exercised that method was one which he had participated in the development of since 1920. Following the modest and uncertain student years of the 1930’s and the war years of the 1940’s, a wealth of students had found their way to university and college campuses. Some of these, indeed, many of these found their way to the University of Texas. Among them were those who sought out the Department of Pure Mathematics. The number of graduate students in mathematics had increased, allowing Moore, as well as others, the opportunity to regularly oﬀer a sequence of graduate courses each year. Moreover, Moore and others on the faculty of the Department of Pure Mathematics, regularly taught undergraduate courses. In that fashion, a student who ﬁrst encountered Moore in the classroom, could be assured of the opportunity to study again with him each semester, until graduation, whether graduation be at the undergraduate or graduate level. There had arisen a natural source of students for Moore. That source was simply the classes which he taught, and it is clear that the greater number of classes taught by Moore, so was the number of students with whom he dealt greater. A student could know that, once encountering Moore in class, there would thereafter be another class, immediately advanced to the last one under Moore, with which he could again encounter Moore, provided he acquired Moore’s consent to enroll in that class.

125 126 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

The Essence of the Moore Method

If one person were to describe to another an impression of Moore’s teaching method, and if they both had experienced his teaching, they probably would agree on certain characteristics, but disagree on the prevalence, or importance, of others. However, there do appear to be some features of the teaching method which nearly all would recognize as being an ingredient of that method which is now called the “Moore” method. The importance attached to each characteristic is as much a function of the student, or of the particular class of which he was a student, as it is of the method. Much of the success of the method depends upon the personality of the teacher, and much of the success depends upon the setting in which the method is practiced. Though that can be said of almost any teaching style, it is perhaps true that some methods offer so little that the personality of the teacher, or the setting of the teaching, is of little effect on the success of the method. Both are of extreme importance in the Moore method. Fundamental to Moore’s method of teaching is the purpose of causing the student to develop his powers at rational thought. The means chosen by Moore was that of posing critically chosen problems, of a mathematical nature, before the students. The choice of the nature of the problem was of extreme importance and was determined by the capability of the class and Moore’s evaluation of their determination. The problem, or question, raised for their consideration needed to be outside their past experience but within their capability to develop the mental procedures or techniques to settle the question. Moore essentially caused students to perform research at their own level. That approach to teaching caused an immediate obligation on the nature and behavior of the class. Obviously, if several of the class were more advanced, in terms of their knowledge of mathematical results and techniques, than were others, then questions posed would sometimes be more accessible to some in the class, on the basis of prior knowledge. Questions raised by Moore were offered in such a fashion that the student would not be informed as to whether the stated conjecture were true. It would be the burden of the student to decide whether to attempt to prove a statement true or describe a counter-example to establish the conjecture false. A student in the course, who had earlier been exposed to the conjecture in some other course, would have some advantage over others in the class and, upon settling the question quickly or by indicating the solution, thereby deprive others in the class of a mental development gained by attacking the problem without such prior information. Thus it developed that Moore would interview those students who wished to enroll in his post calculus courses. If he determined that they “knew too much” he would not permit their taking his course. This was designed to 127 prevent any student from having a built-in advantage of information. Once enrolled, the student was quickly and emphatically instructed that his effort was to be his own; no source books were to be used and no other person was to be consulted regarding the problems posed for the class to settle. A student was intentionally placed in a class of others who were his peers as regards mathematical information. He was presented with mathematical questions to settle and his attempts to settle those questions were his own; no outside help would be allowed. Another fundamental aspect of the method was that of student presentation. Moore would call upon each student, in turn, to present his argument for a particular question which had been raised earlier. A student who claimed a solution made his presentation before Moore and the remainder of the class. Others in the class “refereed” the student’s argument. Those who had spent some hours labor on the problem without success would often be critical and rather jealous referees, wishing that they had succeeded. Those who felt they also had proofs would referee critically, hoping to find error so that they might have opportunity of presentation of their work. If the student offered a successful presentation, he received immediate approval by Moore and others in the class. Though not much might be said, it would be clear that his standing was enhanced. If he had made a logical error in his argument, and had not been able to overcome -t upon its discovery, he would learn in a painful fashion, for it would have been visible to all. If Moore deemed the progress sound enough before the difficulty arose, he often would tell the student, “see if you can fix it up for next time.” In this fashion he developed a feeling of ownership of the problem, on the part of the student, cultivating pride of accomplishment and fostering competition among students. The progress of the class was at the pace of the class. Of course, Moore could alter that pace with two techniques. He could call first on those students who he felt would have obtained correct arguments. This would cause more material to be covered. Or he could systematically call on those first who he felt had little chance to have accomplished a correct argument. Their incorrect efforts, incomplete proofs, and resulting discussion stemming from difficulties in those arguments would slow the pace of the class. Consistent throughout the treatment of the post calculus courses, as taught by Moore, was his capability to raise questions in such a fashion as to not suggest anything regarding the manner in which the problem should be approached. That same impartial treatment was affected by Moore as he observed a student’s effort at proof. Even though the student might be committed to a line of reasoning doomed to failure, Moore would not suggest this by his demeanor or lack of attention. Patience of any instructor, who employs the Moore method, is of paramount importance. 128 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

Moore demanded alert attention among the students in his class. This he partially accomplished by way of his stance and attentive behavior. How- ever, he would eject from his class any student whose behavior interfered with the opportunity of others in the class to follow the proceedings of the class. Moore spent much time evaluating members of his classes. A student of the early 1950’s recalls

. . . Moore was a very talented man who was dedicated to teaching and spent a lot of his time thinking about his students and was brilliant enough to do the right thing. That is, he thought about these people in his classes. He spent an enormous amount of time thinking about people in his classes and what was going on. I was a real good student in high school. I walked into classes and I listened to what people had to say, and I made all A’s, and I hated every minute of it. I was a dead-end kid. I spent all of my time raising hell. And I found nothing intellectual to challenge me in the least until I walked into Moore’s class as a freshman at the University of Texas and for the ﬁrst time, found something that was being presented to me that I could thoroughly and completely understand. I was provided with challenges to try to do things, the likes of which, intellectually, I had never seen in my life. Challenges that were just barely within my grasp - time after time after time. If he“d given me problems that were just a little bit harder, I would have said, “The hell with him!” If he” d given me problems that were just a little bit easier, I would have said, “The hell with him!” There were consistently things in that class that were just barely beyond my reach. Now, it may have been just a coincidence that he hit me that close, but I doubt it. I just feel like the man knew what he was doing . . .1

Included in his day-to-day evaluation of each class was a decision as to how he would pace that class. Would he call on the weakest student, hoping for incorrect statements and glaring errors in logic, or would he call upon the ablest, so as to get certain questions behind them? Would he offer a sequence of elementary problems, the solution of which would be designed toward developing the procedures and techniques to settle a difficult question, or would he offer straightforwardly the difficult question with no helpful lemmas? He varied his method to fit the students, keeping foremost

1Personal conversation with W. S. Mahavier, Atlanta, Georgia, February 17, 1971. 129 his own evaluation as to which approach would provide most accomplishment. In one class, he had consistently called upon one man who he felt was one of the two weakest in the class. He gained, with that approach, a succession of mathematical and logical errors which could be examined by the offending student and all members of the class. Toward the end of the fall semester, Moore raised a question for that class which afterward, he realized was quite difficult. Pondering the problem, he saw that if one were to settle another question, the first would follow quickly. But he doubted that anyone in the class would come upon a statement of the intermediate question. The very next class meeting after raising that difficult question, the weak student, who Moore had so often called upon, claimed an argument for it. Allowing him to go to the board, Moore was surprised to observe his stating precisely that needed intermediate lemma and then proceed well into the argument, finishing it the next period. Following that one spec- tacular performance though, the student “lapsed into his original state.” 2 Indeed, the practice was as before; Moore would call upon that student, only to have him give fallacious arguments with bad logic. Several months passed in that manner. Finally, the student began to perform better, until he was among the best, if not the best, in the class. Perhaps it was that Moore had detected something deeper than a predictable presentation filled with nice errors. Perhaps he detected dormant talent and pursued it until it flourished. Any such pursuit came at the expense of devoting similar classroom time and attention to other students. Thus, some decision was made by Moore to “pick-on” that particular student instead of focusing on another or others in the class. An example in the other extreme comes from that same period. One student seemed to have settled, without fail, whatever questions had been raised the preceding class period. Moore consistently called on that student. Even though he might call on others before he called on that student, the shortness of time between classes often had not allowed others ample opportunity to settle the questions. Calling on that particular student was a decision on Moore’s part consistently to allow that student opportunity to settle a conjecture before others in the class had settled it. Resulting was something similar to a lecture course, with that student lecturing. At the end of that year, two of the students asked Moore to let them drop out of his course sequence for a year, returning so as to be no longer competing in class with the student Moore later was to refer to as Mr. “S.” Moore allowed two promising mathematicians to depart, with there being natural uncertainty of their return. Why would he consistently allow Mr. “S” the

2Film: Challenge in the Classroom, The Mathematical Association of America, filmed in 1964-1965, Austin, Texas. 130 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE” opportunity to present his arguments before others had completed their efforts on a question? Perhaps he felt that Mr. “S” had such unusual ability that he would pursue that development at the expense of others. Perhaps he felt the others in the class were good enough to grapple with competition, as they would as professional mathematicians, on a survival- of-the-fittest basis. Perhaps he just simply enjoyed seeing Mr. “S” progress and refused to restrain his own desire to observe Mr. “S” perform. Perhaps he recognized the ability of Mr. “S,” but felt Mr. “S,” as he had G. T. Whyburn years earlier, needed to be utterly convinced that he had superb mathematical ability. Or perhaps it was an accumulation of all those things, and others. It is that facet of Moore’s teaching that depends upon the man as opposed to the method. A list of procedures and techniques can suggest the method. But how does one describe the criteria to determine when to allow one student opportunity to dominate a class and when to “retard” a class to allow dormant talent of a student the opportunity to flourish? Though criteria can be stated which allow an atmosphere for student competition, how does one describe those personal qualities which were such that Moore could cause a student to work and think harder than he ever had before? Criteria which characterize the Moore method of teaching include:

1. The fundamental purpose: that of causing a student to develop his power at rational thought.

2. Collecting the students in classes with common mathematical knowledge, striking from membership of a class any student whose knowledge is too advanced over others in the class.

3. Causing students to perform research at their level by confronting the class with impartially posed questions and conjectures which are at the limits of their capability.

4. Allowing no collective eﬀort on the part of the students inside or outside of class, and allowing the use of no source material.

5. Calling on students for presentation of the-r eﬀorts at settling questions raised, allowing a feeling of “ownership” of a theorem to develop.

6. Fostering competition between students over the settling of questions raised.

7. Developing skills of critical analysis among the class by burdening students therein with the assignment of “refereeing” an argument presented. 131

8. Pacing the class to best develop the talent among its membership. 9. Burdening the instructor with the obligation to not assist, yet respond to incorrect statements, or discussions arising from incorrect statements, with immediate examples or logically sound propositions to make clear the objection or understanding. Surely, other characteristics would occur to those who were ever in a class taught by Robert Lee Moore. Teaching by the Moore method causes extreme obligation to fall on the teacher. Moore met that obligation and, with him, the method was eminently successful. Others, who wish to employ the method, fail to meet the obligations they thus incur and they fail miserably. Still others partly succeed, but weaken the application of the method and reduce the dividend from it. Still others have not the patience to attempt that method, but do so in any case, with a result that practices far removed from the Moore method go by that name. What are some of these obligations that one should accept if he is to teach by the Moore method? One of the obligations that Moore accepted was to do that which was necessary to cause the student to develop his power at rational thought. That single decision led him to hold firm to positions that caused hurt feelings, misunderstandings, and philosophical differences with colleagues. For instance, his determination to close his classes to those students he felt were too knowledgeable, caused unhappiness among those students who wished to enroll in his course. Indeed, as they carried their unhappy complaint to other faculty, or administrators, those people felt need to defend or attack Moore’s policy, depending upon their own understanding of it and agreement with it. That same insistence of Moore’s, to close his classes to those he considered too knowledgeable, caused others to alter their own teaching. That is, a faculty member, who taught courses which ere prerequisite to one of Moore’s courses, taught his own course knowing that introduction of certain subject matter would strike students of his class from taking courses later with Moore. Some faculty were not unhappy with this arrangement, feeling that Moore was on sound academic ground. Others though, felt that Moore was dictating the content of their course to them, by closing his classes to students who had seen certain topics or concepts introduced beforehand. A rather classic academic difference would grow between faculty members at the University of Texas; some would claim the privilege of teaching however they pleased and would surely grant that privilege to any other faculty member. However, some of the others, while having no formal restriction as to how nor what they taught, found their students would not gain admittance to Moore’s classes if certain topics were taught them “too early.” An indirect pressure arose: Each faculty member, in teaching his own class, 132 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE” would grapple with the introduction of a fundamental concept. Would that strike a student later from membership in one of Moore’s classes? Moore would not compromise his position, being willing to sacrifice the teaching of some students so that those he taught would first meet new concepts in the manner he wished. Other aspects of Moore’s teaching caused difficulty with other faculty. Moore simply would not allow a student to seek assistance elsewhere, either from other people or other source books. In those instances that Moore became convinced a student had violated either of Moore’s standards for participation in his class, Moore would eject him from class. Other faculty felt it very reasonable that a student should consult with others, as well as referring to source books. That philosophical difference was apparent in many forms. If Moore ejected a student for cheating or resorting to other sources, often a fellow faculty member or the chairman would also defend or condemn Moore’s action. Students, taken by Moore’s approach, often would exhibit strong disdain to other faculty and other students whose philosophy was of the “other school.” Perhaps no other form of that disdain was more visible than that of a student, captured by Moore’s philosophy, who would regally ignore in-class proofs given by his teacher, choosing instead to prove each presented theorem for himself. The faculty member whose own training was not in sympathy with Moore’s approach, often felt bitter concerning the arrogance of Moore’s students. Moore was not the only faculty member who drew that sort of reaction. en the Department of Pure Mathematics moved into Benedict Hall, it was assured of certain office and classroom assignments. There was concern among them that, by moving into the same building with the Department of Applied Mathematics and Astronomy, the formal separation of the two departments would be difficult to continue. A portion of the agreement reached to facilitate the move was that of assuring the faculty of the Depart- ment of Pure Mathematics the area which was the third floor of Benedict Hall. There would develop before long an informal definition of “third floor” and “second floor” mathematics with the former representing the “Pure” and the latter the “Applied” groups. It would be with pride that faculty of the third floor group would return from distant meetings, recounting such statements, made there by others, as: “Those third floor people from Texas don’t know anything, but they can do everything.” Indeed, that same arrogance on the part of students, which some faculty would find offensive, was developed purposely by Moore and other “third floor” faculty. The student needed to believe that, if he only understood the definitions and the statement of the conjecture then he could settle the question. More- over, he needed to remain somewhat ignorant, so that his own attack of the question would be unfettered, and not restricted by another person’s 133 approach to the problem. Many related difficulties arose. Moore held fast to his position, teaching in that fashion which was so effective. His refusal to compromise his method disturbed both faculty and students who were not in sympathy with his method. The faculty of the Department of Applied Mathematics and Astronomy had years earlier lost much of its “applied” character. First, students were allowed to take the mathematics in either department to satisfy degree requirements, suggesting there was little distinction between the offerings of the two departments. The interests of the faculty of the De- partment of Applied Mathematics had long since been not appropriately described. Indeed, as Professor Vandiver moved into that department, its offerings could no longer be correctly called only “applied.” As new faculty were added, responding to increasing student enrollment, many were hired who considered themselves research and non-applied mathematicians. Re- sentment grew between the two faculties, resulting in substantial efforts to alter the arrangement of there being two faculties and two departments of mathematics. By the early 1950’s, C. P. Boner of the Physics Department was holding the office of Dean of the College of Arts and Sciences. Boner had many years of experience with the Department of Mathematics, even having served as President of the mathematical honor society, the Pentagram, in 1921. He knew Moore well and had definite philosophies as to the advantages and disadvantages of there being two departments of mathematics. He, after much discussion with many faculty members, at Texas and from away, moved to have the two departments joined. His efforts were successful. Noted in the 1953-1954 catalog is the statement:

The Department of Applied Mathematics and Astronomy and the Department of Pure Mathematics were combined at the beginning of the 1953-1954 Long Session.

The faculty at that time were Professor Emeritus M. B. Porter; Distin- guished Professors Moore, Vandiver; Professors Cooper, Craig, Ettlinger, Wall; Associate Professors Batchelder, , Lane, Lubben; Assistant Professors Edmonds, Guy, G. H. Porter, Prouse. That joining of the two departments coincided exactly with the first year that Robert Lee Moore began his “modified service.” He had experienced his 70th birthday during the long session of 1952-1953. These who felt that mathematics at Texas should not be so strongly influenced by Moore were taking steps to diminish that influence. By combining the two departments, only one Budget Council (the departmental executive committee) would exist and it could be expected that the influence of the “applied” faculty 134 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE” would dominate, if only by the press of numbers. During the fall of 1953, the first year Moore was experiencing “modified service,” a petition from the Faculty Council to the Board of Regents was made, asking for a revision of the rules and regulations of the Board in regard to Modified Service. That matter had originated at the October meeting of the Faculty Council when Dean Boner ’sought the advice of the Faculty in regard to questions which had arisen in the administration of the modified service regulations.”3 Resulting from that request was a committee, appointed by the President of the University, to study the matter and report back to the Faculty Council.4 The practice at the University of Texas was such that actions taken by the Faculty Council were circulated to the General Faculty. There followed a ten-day protest period. If as many as eleven faculty registered formal written protest, the matter would be brought before the General Faculty in formal session. Otherwise, the action of the Faculty Council would be final. In part, the action taken by the Faculty Council was that of recom- mending that:

No person on modiﬁed service shall carry more than one-half the regular duty of a full-time employee. This work may consist of classroom teaching, conference instruction, research, directing research projects, or other departmental or university duties, but in no case shall the sum of these responsibilities exceed one-half of regular full-time duty.5

Protesting that action were eleven faculty members. Included in that number were Moore, Ettlinger, Wall, Lane, and Lubben, all of the “third ﬂoor” segment of the Department of Mathematics. Moores protest read:

I protest the action of the Faculty Council taken on January 18, 1954, concerning modiﬁed service, and I request that this matter be referred to the General Faculty. I regard Item I as absolutely inexcusable and indefensible.6

3Minutes of the General Faculty, University of Texas, Austin, Texas, March 9, 1954, p. 6285. 4The membership of that committee was: Mr. M. V. Barton, Mr. H. V. Craig, Mr. H. R. Henze, Mr. H. M. MacDonald, Mr. Carson McGuire, Mr. A. W. Nolle, Mr. H. H. Otto, Mr. O. W. Reinmuth, Mr. Robert P. Wagner, Mr. F. C. Wegener, Mr. J. A. White, Mr. Orville Wyss, and Mr. David L. Miller, Chairman. 5Minutes of the General Faculty, University of Texas, Austin, Texas, March 9, 1954, p. 6286. 6Documents and Minutes distributed among the faculty members of the University of Texas, February 24, 1954, p. 6255. 135

Resulting from those eleven protests was a meeting of the General Fac- ulty on Tuesday, March 9. A record of the events is as follows:

The Secretary moved the adoption of the petition of the Faculty Council as circulated (G.F. 6253-54). The motion was seconded by Mr. David Miller. Mr. R. L. Moore moved that the petition be taken up item by item and the motion was seconded. Mr. Law stated that the report hung together very well and thought that it would be unfortunate to limit discussion to each particular item when there might be other provisions of the report which should be discussed in connection with that item. President Wilson ruled that in the discussion of any particular item discussion of other items would be in order if pertinent. The motion to consider the petition item by item was then adopted without dissent. President Wilson recognized Mr. David Miller, Chairman of the Special Committee which had presented the original report to the Faculty Council. Mr. Miller recalled that the discussion of a retirement system and modiﬁed service began in 1935 and was put into eﬀect in 1937. He then read the following excerpt from the Brogan report of 1935 (D&P 844) as indicating the intentions of the Faculty at that time:

Every voting member of the General Faculty or Med- ical Faculty . . . shall be placed on half of his previous duty beginning with the ﬁscal year that follows his seventieth birthday . . .

In 1949 Mr. Simmons, Chairman of a General Faculty committee reported:

The Committee believes that, in general, the assignment of work to a person on modiﬁed service should not exceed one-half his load under standard service.

Mr. Miller pointed out that the original intention was to have everybody completely retired at 70 but experience had shown that it might be wise to retain members over 70 provided they were capable of performing their duties. We were rapidly ap- proaching the time when the beneﬁts under the teacher retirement system would be one-half of the salary of a person at age 70 when he went on modiﬁed service. When this time arrived 136 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

it would be possible to carry out the original intention of complete retirement at age 70. However, the committee did not think compulsory retirement at age 70 was a wise thing. There were some people who could teach beyond that age. Therefore, the committee had proposed that the budget councils of the departments should recommend to the administration whether or not they thought that the person in question was capable of continuing to teach. These were merely recommendations to the administration and to the Regents so that they would have a basis of determining whether or not a particular individual should be completely retired or kept on modiﬁed service. Mr. Miller stated that there would be other questions which would come up later and that he would discuss them at that time. He, thereupon, moved the adoption of Item I, reading as follows:

The motion was seconded. Mr. R. L. Moore then addressed the faculty as follows:

Item I is not very clearly stated but it seems to me that he main issue with respect to it is too important, and the time we have left this afternoon is too limited, to permit of a discussion of all of its possible interpretations. Let us concentrate on one simple proposal which I hope every one here will agree is a part, if not the whole, of whatever there is in it that is enforceable, namely the simple proposal that no one on modified service shall teach more than two three semester hour courses in any one semester. Is this a university or is it a union shop, its so-called faculty consisting of mere employees, each doing a specified amount of work for a specified wage, each fearful that if one man does more than he is paid to do there will not be enough work left for the others to do, a union shop where, according to the Faculty 137

Council minutes, one member says “the question is whether you could put more than one-half time work on a man who got only one-half time pay, even if he wanted more,” as though teaching a class that he is not specifically paid to teach is bound to be a burden and if a man asks that such a burden be “put” on him then either he is just trying to appear to be accommo- dating, but is secretly hoping that his request will not be granted, or he is so far gone mentally that he does not realize what he is doing and therefore should not be allowed to teach at all? A union shop where the chairman of a committee brings in a report containing an item that would require that no one on modified service shall be allowed to teach more than half time and apparently is strongly in favor of that requirement but, when asked by the president whether the committee had given any attention to the minimum load of persons on modified service, shows little or no concern as to whether or not these people should be required, or even expected, to do anything at ali worth while, saying that if a person is “not capable of teaching he could be assigned to research or something of that nature?” If a person on modified service who had formerly taught full time is no longer capable of teaching at all, I wonder what sort of research, if any, he would be capable of carrying on. I hope that most members of this faculty do not have this union shop attitude. If a man on modified service does more than he is paid for and his work is sufficiently good and he is doing it with no expec- tation whatsoever of any extra pay then, instead of condemning, should not the administration approve of the extra work he is doing on the ground that, by doing more than he is paid for, he compensates to some extent for those who are doing less than they are paid for and in that way makes some contribution towards making the modified service system pay its own way? Now I hope that no one here will interpret anything that I have said or that I am going to say as implying that every one on modified service, or even that 138 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

every one with full-time status, should teach at least two courses. I never said, and I never meant to imply, and I do not believe, any such thing. There are many types of teachers, subjects and students and I think it is quite possible that some individual on full- time might offer only two courses (or possibly only one course, or even none at all) and contribute much more to the advancement of his subject and be of much more value to his students and the University than he would have been if some regulation had forced him to offer more. For example I should think this might be so in the case of some one who is carrying on an important investigation with the assistance of numerous students each of whom is attacking some aspect of the problem and having frequent conferences with him concerning it. I do not think there should be any hard and fast rule that fixes a specific number an provides that no one shall teach less than that many courses or that provides that no one shall teach more than that may courses. According to the Faculty Council Minutes Dean Haskew said “Regardless of the general understanding that a person on modified service was to teach not more than one-half time and be paid for that, we found sympathetic departments and sympathetic officials making and accepting exceptions up and down the line” and Dean Boner said appeals had come to him “from departments in which courses that some youngster would like to give were being taught by persons on modified service. When asked why they would not recommend a change, the departments said they simply could not because of courtesy to the older man. We were letting the person on modified service continue to teach the course in order to save the embarrassment of telling him that he had to give it up.” Now if a so-called youngster - I prefer another term if a young man wants to give a course taught by a man on modified service and the older man is not carrying more than half time work then the provisions of Item I would not apply to this older man and would not force him to surrender any of his courses to the 139 younger one or to anyone else. The number of en on modified service who are giving more than two courses is extremely small and I doubt whether there is any young man in this university who is both desirous and capable of giving a course which is now given by anyone in that small group. But whether or not the older persons referred to by Dean Boner were giving more than two courses, and whether or not they were on modified service, if the so-called youngsters in question were more capable of giving these courses than were these older persons and the departments asked that such a regulation as Item I be put into effect for the sole purpose of saving them the embarrassment of doing their duty in the mater and the dean also asked that such a regulation be adopted solely to save himself embarrassment, both he and they being much concerned about embarrassment to themselves but callously indifferent to any damage such a regulation might cause either now or in the future, either in the same department or in some other department in which there is no such qualified “youngster” then I would like to ask the dean and these departments whether they are really thinking of the best interests of the University and its students or whether they aren’t just thinking of themselves? As to Dean Haskew’s statement, if his words mean that there has been a general understanding that no one on modified service shall be permitted to teach more than one-half time, I have been here much longer than Dean Haskew and, as far as I remember, before this committee was appointed I never heard that there was any such understanding or that anyone had eve proposed that there by any such understanding. In fact, a member of the administration immediately preceding this one indicated to me that no such regulation existed and made commendatory remarks concerning certain people on modified service, or about to go on modified service, who had done, were doing, or were expecting to do, more than they were required to do. He also indicated that modified service, as it now exists, is a good thing and that he hoped that it 140 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

would not be tampered with. As to whether Dean Haskew is right in saying that there have been exceptions “up and down the line” to this so-called general understanding, I have talked to numerous individuals on modified service, to chair- men of departments and to others who I thought might be in a position to know about this situation and if there are more than two people on modified service who are teaching more than two courses this semester I have failed to uncover that fact. I understand that there is one who is giving only three courses but I have been assured that if there ever was any departmental problem concerning him it has already been resolved. That apparently leaves, as the only present target of Item I, a man who is giving five courses. I have full authority to speak for him. I will call him Profes- sor X. He has felt in the past, and he still feels, that he can contribute much more to the advancement of his subject through his own researches and those of his students with the discovery and development of some of whose abilities he has had, or may in the future have, something to do than he could possibly have contributed by his own researches alone, even if he had been entirely free to devote all of his time to research and done no teaching whatsoever. With this in mind he has been giving a sequence of five courses. The first one, 613, is one section of a sophomore course which he has been giving partly, but not wholly, because he felt that he might discover some new material there. The next course, 624, is only one of a number of sequels to 613, the other possible sequels being given by others. The next one, 688, is only one of a number of possible sequels to 624, and so forth through 689 and 690. No course in this sequence beyond the first one has ever been given here by anyone else. At the present time, in each of these courses except, of course, the last one, there is prospective material for the next one and thus there is a reason for continuing to give all of them. Without devoting a lot of time to a description of the unusual manner 141 in which these courses are given it would be difficult or impossible to make clear all the ways in which it would be unsatisfactory to have one man give one or more of them and have another man give the others. I would not want to undertake to explain it here this afternoon and I do not think you would want to listen to me if I did undertake it. The first Ph.D. with the dissertation under the supervision of Professor X was conferred in 1916. The last one so far was conferred in 1953. The interval from May or June, 1916, to May or June, 1924, will be referred to as an eight year period. So will the one from May or June, 1917 to May or June, 1925 and so forth. The group of all students who received Ph.D. degrees with dissertations under the supervision of Professor X at any time during the period from l919 to 1927, for example, will be referred to as the l919 to 1927 group and similarly for other periods and groups. In the entire interval from 1916 to 1953 there were 30 such groups there being of course many overlappings. If Professor X were forced to pick out one of these groups of students and say that it was the best one I think he would say the last one, the 1945-53 group. He would be inclined to say that in spite of the fact that the 1922-30 group included, among others, both the man who is now serving as President of the Amer- ican Mathematical Society for the two years 1953 and 1954 and the one who has been elected to serve in that capacity during the two years 1955 and 1956. The total number of people who have received Ph.D. degrees with dissertations under the supervision of Professor X is 26. Nine of these belong to the 1945- 53 group. The total number of these 26 people who are now on the faculties of members of the Associa- tion of American Universities is 13, and six of these belong to the 1945-53 group. Of these six the one who received his degree in 1953 is the only one who does not have professorial rank. The universities at which these six are located are The University of Virginia, Syracuse University, The University of Pennsylvania, The University of Michigan, The University of Wis- 142 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

consin, and The University of Minnesota. Seven of the twenty-six are now members of the Council of the American Mathematical Society, a body which in 1953 had a total membership of 48. Of these 7, two belong to the 1945-53 group. There are 5 of the twenty-six who are either starred in American Men of Science or listed in the “Distin- guished Group” in the recently published Leaders in American Science. Of these 5, two are in the 1945-53 group. So far as I have discovered, the total number of all people who have received Ph.D.’s at this Uni- versity with dissertations under the supervision of any present member of this faculty outside of the Depart- ment of Mathematics and who are either so starred or so listed is 2. One of these two received his degree in Zoology in 1915 and the other one received his in Organic Chemistry in 1941 so that neither of them received his degree in the period from 1945-53.

The President interrupted the speaker stating that the Secre- tary had called his attention to the faculty rule limiting the discussion of a speaker to ten minutes on any one proposition and that this period had been exceeded by the speaker. Mr. Manuel moved that Mr. Moore be given the privilege of extending his remarks, and the motion was seconded and adopted without dissent. Mr. Moore continued as follows:

Making use of American Men of Science and other sources of information, I have consumed much time trying to find out whether or not any one of the approximately 65 people who have taken Ph.D. degrees in Physics at this University is starred in American Men of Science or is on the “Distinguished List” in Leaders in American Men of Science, or is a member of the Council of the American Physical Society or is on the faculty of any member of the Association of American Universities other than the University of Texas. So far I have found no one fulfilling any one of these conditions except the last one and the only ore I have discovered who fulfills that one is a professor at The University of Illinois and he is a professor of 143

Electrical Engineering, not of Physics.7 In 1948-49, in 613, the first course in the above mentioned sequence of five, there were 4 undergraduate students who were majoring in Physics. In 1949-50 all of them were in 624 and they were joined in that course by a student also majoring in Physics who had graduated at the California Institute of Technology. Of these four undergraduates, one graduated in 1951 and continued in Physics. The three others graduated in 1951 (one with highest honors) and are now all enrolled in 690, the next most advanced course in that sequence of five. The graduate major in Physics, who was with them in 64 in 1949-50, took no more courses in Physics after that year though he wrote a thesis for a Master’s degree in Physics and received that degree in 1951 or 1952. In 1953 he received a Ph.D. degree with Pure Mathematics as major and Physics as minor subject. He went to The University of Minnesota as an instructor in mathematics at an initial salary of forty-five hundred dollars. I consider him to be one of the very best of the entire 26. Professor X would like very much to have the opportunity to try to make the eight year period from 1954 to 1962 at least as good as the one ending in 1953. But I think that it certainly will not be at all possible for him to do it if Item I is put into effect. I hope that no one will move to amend Item I. At the Faculty Council meeting at which this item was being debated, Dean Brogan proposed an amendment providing, in part, that modified service shall be one- half the regular duty of a full-time employee except as the appropriate officials and President may recommend departure therefrom. According to the Faculty Council minutes, Dean Boner urged the Council to vote “No” on this amendment and, referring to Item I, he stated that any departure from the rule of the committee that he would approve would have to have

7Recall that Boner, the Dean of the College of Arts and Sciences, who had been instrumental in the formation of the single Department of Mathematics, was a physicist by academic training, and had been the instigator of the efforts to have a new definition of modified service. 144 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

the unanimous approval of the entire voting faculty of the department and the approval of the dean. In view of this statement by Dean Boner is it not clear that it would be entirely fruitless to adopt any amendment providing for exceptions? But I would be strongly opposed to this amendment even if Dean Boner had not made clear his attitude concerning exceptions. There is a principle involved here. Should anyone have to ask the permission of the administration every time he wishes to do something that is not prescribed? The The fact that a thing is unusual does not brand it as being irregular and I hope that this faculty does not approve of any motion which even suggests that doing more than one is required, or expected, to do is irregular and that anyone who wishes to do it must ﬁrst get the permission of the administration. Has uniformity become a virtue and has every departure from it become an oﬀense to be tolerated only by special permission in each individual instance?

Mr. Ettlinger addressed the faculty as follows:

The recommendation contained in Item I of the action of the Faculty Council which is now before us is open to at least three objections. First, it is based on the principle that no employee on modiﬁed service shall render more than one-half time duty to the University. The University of Texas is at present committed to retirement at age seventy, either completely or on modiﬁed service. The usual situation is that the teaching member is content to render one-half of regular full-time duty. Many full-time faculty members render far more than their full-time responsibility requires. The principle of “Thou shalt not” render more than you are paid for, is extremely undesirable. Who will hold the stop watch on a faculty member engaged in his own research, or in conference with a graduate or undergraduate student, and tell him “just seven more minutes” and you must cease and desist or you will be violating the work rules of the University of 145

Texas. Second, the principle embodied in Item I is completely out of line with Item 2. In Item 2, the basis is definitely on the competence of the department to administer its affairs with the welfare of our students, of the faculty and of the University as the guiding principle. But in Item I, the department is adjudged incompetent to exercise good judgment to protect the younger faculty members in their desire to offer courses in their fields, and also to protect students from incompetent teaching. But who will install the time clock in the department, who will patrol the laboratories, who will take the reading of the light meters.? Will quarterly reports be made and to whom as to the precise expenditure of time? If you urge that bu- reaucracy will not make its appearance, then it may be proper to as, how will it be determined whether “classroom teaching, conference instruction, research, directing research projects, or other departmental or University duties add up to two-thirds of five-fourths of regular full-time duty and hence constitute a vi- olation of the work code for modified service. This item just does not add up and this General Faculty should not recommend a regulation which is unenforceable or without meaning. Third, this regulation is not compatible with the ideals of a University of the first class. Alfred North Whitehead served brilliantly as a faculty member of Harvard University from the age of sixty until close to eighty-five, without any regulation circumscribing his intellectual activities. Cannot the University of Texas afford the luxury of an Al- fred North Whitehead? For these three reasons, that the regulation will prove to be unenforceable, that it is undesirable since the department can best protect the interest of its own faculty members as well as its students, and because it will handicap unnecessarily the University of Texas, I favor the deletion of Item I, of the report.8

8Minutes of the General Faculty, University of Texas, Austin, Texas, March 9, 1954, pp. 6286-6290. 146 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

Following further discussion, Item I was put to the faculty and defeated by a vote of 82 noes and 62 yeas. Immediately thereafter, Items II, III, and IV were taken up and adopted.

ITEM II

There shall be no general rule regarding the teaching of required courses or sections by teachers on modiﬁed service. Rather the departmental teaching staﬀ of each department concerned shall determine this matter each time a course schedule is composed.

ITEM III

Persons on modiﬁed service shall not teach during a Summer Session except to meet unusual departmental needs as determined by the Budget Council of the department concerned. In case of such emergency, the Budget Council, with the consent of the person on modiﬁed service, may make a special request to the dean or other appropriate administrator for the employment of that person during the Summer Session.

ITEM IV

During the fiscal year in which an employee’s 70th birthday occurs and before the budget is prepared for the following year and each year thereafter until the employee is fully retired, the members of the Budget Council of his department, or, in case he belongs to no department, the Administrative Council, shall report to the dean, or other directly superior administrator, concerning the overall fitness of the person on modified service to fulfill his duties. This annual report must state that its authors either believe or do not believe the person in question is capable of carrying his modified service duties.9

Discussion had begun on Item V, but lateness of the afternoon caused adjournment before action was taken. The General Faculty next convened on the afternoon of March 23 and again took up the question of Item V.

ITEM V

9Minutes of the General Faculty, University of Texas, Austin, Texas, March 9, 1954, pp . 6296-6294 . P 147

At the beginning of the fiscal year following the fiscal year in which the 75th birthday of an employee on modified service occurs, the employee shall be retired automatically. However, in exceptional cases, if the employee is considered by the Budget Council of the department concerned to be competent to fulfill his duties, or, if the employee is not a member of any department, if the Administrative Council considers him competent to fulfill his duties, said Budget Council or the Administrative Council may petition the appropriate dean or other appropriate administrative officer to continue the employee on modified service. Such a petition must be made annually thereafter if the person concerned is to be continued on modified service.10

Miller the secretary oﬀered comments, including:

The basic reason for the recommendation of the committee was to have a clarification of the duties of members on modified service. It had been the hope of the committee to safeguard the modified service program and to do it in a reasonable way. The alternative to modified service was complete retirement at 70. The committee did not think that the Faculty wanted that, but that the Faculty wanted an administration of the modified service program so that people who were qualified to teach might be permitted to continue to teach. The proposals of the committee were not directed at any one person and as far as he knew no names had been mentioned at the committee meetings. The committee wanted it to be possible for those people who were competent to continue teaching to be permitted to do so, but they also wanted the University and the students to be protected. The proposed recommendations would require the budget councils to furnish the information upon which the President and the Board of Regents could act and thus avoid any general rule requiring retirement at 70 or 75. . . . In Recommendation V, when a person reached 75, his services would be discontinued unless the budget council believed that he was competent to carry on his modified service duties. In that way anyone who was incompetent to carry on would be retired. The rule would be applied to everyone alike–to the members of

10Minutes of the General Faculty, University of Texas, Austin, Texas, March 23, 1954, p. 6295. 148 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

the committee and the members of the Faculty Council as well as to other members of the General Faculty. The committee had in mind not to abuse any one. It was trying to get a sensible, rational program to save for the University the good men of the University –the people of 70 and 75 who were capable of carrying on their duties.11 Following Miller’s comments, Moore read the following: A report of a Special Committee on Modified Service and its Relation to the State Retirement System was adopted by the Faculty Council on December 19, 1949. This report contains the following statement: “When originally adopted the Modified Service System was considered a sort of interim arrangement to be discontinued when a satisfactory retirement system had been authorized and in operation long enough to produce adequate retirement benefits for members coming to retirement age. This latter ha, by no means been achieved. By the adoption in 1948 of recommendations made by the Committee of which Dean Brogan was Chairman, approval was given of a rule which in practical effect assures the continuance of the present modified service system for an indefinite period. As our experience with it has developed, there are many who feel that it has certain intrinsic merit which will justify its permanent retention in some form or other beyond the time, if ever, when we have a fully adequate retirement system.” I wish to proceed on the assumption that, as presently administered, this system does have such intrinsic merit and that, regardless of the original occasion for its inception, it is for the best interests of the University and its students and the advancement of learning that it be continued, provided it is continued in such a way as to best serve those interests, not merely for the direct personal benefit either of those faculty members who are on modified service or of the nonstudent faculty members who are not. I certainly would not be in favor of adopting a regulation requiring that a faculty member over 70 shall be retained on modified service if the only way to retain him is by having him teach classes that he is not capable of teaching properly. Such a regulation might conceivably be for the direct personal benefit of

11Minutes of the General Faculty, University of Texas, Austin, Texas, March 23, 1954, pp. 6295-6297. 149 some faculty members and their families but I certainly do not think it would be for the best interests at the University and its students and the advancement of learning. But neither am I in favor of a regulation requiring that if a man over 75 shows by his actual performance that he is still outstandingly capable of continuing to teach and inspire and contribute to the advancement of his subject, he shall nevertheless be replaced either gradually or otherwise by some younger man who no one has any good reason to think could come anywhere near matching either his past or his future performance. Such a regulation might be for the direct personal benefit of some younger faculty members but does any one really think it would be for the best interests of the University or its students or the advancement of any field of work? Now if this Committee has planned its program without reference to the possible existence of such outstanding men then it has been planning for a mediocre university, perhaps for a university in which there is such a dead level of mediocrity that whenever a member of its faculty nears the age of seventy then, no matter in what subdivision of what field he is working, there is always a younger man who is just as good (and probably better since he is younger) and who is available and procurable and easily made ready to gradually take over and continue with his work. Does anyone think that really first class men are that plentiful? The force of these objections to Item 5 is not materially lessened by the fact that it contains a provision to the effect that in exceptional cases a budget council may petition the appropriate dean to continue a professor over 75 on modified service provided such a petition is made annually thereafter if the person concerned is to be continued on modified service. Who would be willing to be made the subject of such a petition to be renewed annually as indicated and perhaps submitted to a dean who might take the same attitude towards Item 5 as he took towards Item 1 and refuse to make any exception unless it was agreed to by every voting member of the department concerned? If the budget councils, departments and administrative officers perform their plain duty in each individual case, as it arises, there will be no need for any such regulation as Item V. In order to relieve these groups and individuals of the performance of their plain duty with respect to the incapable is this faculty 150 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

going to vote for the adoption of an automatic regulation which would apply, without distinction, both to the capable and the incapable and would make it impossible for even the most capable to continue beyond some arbitrary specified age fixed in advance? Remember the first sentence of Item V reads as follows: “At the beginning of the fiscal ear following the fiscal year in which the seventy-fifth birthday of an employee on modified service occurs, the employee shall be retired automatically.” I hope that if any of you are inclined to vote for the enactment of this automatic regulation you will first give serious consideration to the question whether or not such a vote would really be for the best interests of the University and its students and the advancement of the various fields of learning which is one of the chief purposes for which any first-class university exists.12

After more discussion, Item V was brought to a vote and defeated. Shortly thereafter, Miller stated:

. . . that as the General Faculty had voted down the Faculty Council’s recommendation on Item I, he thought it was necessary to complete the report by including some statement of the duties of the person on modified service. He had a statement which he thought was straightforward and could be readily applied. He, thereupon, moved the adoption of the following statement: “The assigned duties of a person on modified service shall be approximately one-half of his duties while on full-time employment.” After his motion was seconded, Mr. Miller stated that the basic reason for the committee’s recommendation was that the Ad- ministration had been approached on several occasions in regard to the question of teaching loads of persons on modified service. Dean Boner could support the fact that persons from different departments had inquired to find out specifically how teaching loads should be interpreted. Dean Boner had stated that there was no clear-cut rule and had asked for a special committee to study the problem. He believed this statement would help clarify the situation.

12Minutes of the General Faculty, University of Texas, Austin, Texas, March 23, 1954, pp. 6297-6298. 151

Mr. Moore asked what was meant by “assigned duties” and stated that he thought that the wording was too ambiguous. Mr. Miller stated that the individual could consult with as many students as he wanted to and could do whatever research he wanted to but that the duties assigned to him would have to approximate one-half of his duties while on full-time employment. Mr. Moore stated that he considered this an attempt to enact Item I with a different wording. He objected to this matter being brought up at twenty-minutes to six and stated that time was needed to study the statement. Mr. Schoch asked why this statement was proposed as against the regent’s statement of the duties of a person on modified service. Mr. Miller replied that it was his understanding that the Re- gent’s rule was not clear and that the proposed statement would clarify the constituted duties and responsibilities. Mr. Schoch stated that the matter had come up in the Faculty before it went to the Board of Regents the first time. The rule was deliberately phrased as it was. It had been considered carefully and it had been determined that it was the wise course not to be too specific. Mr. Miller stated that the present rule was apparently creat- ing difficulties of interpretation and enforcement, and for that reason the Faculty Council had been asked to submit a new wording. After further discussion as to the meaning of “assigned duties” and as to whether Mr. Miller’s statement was in reality a reordering of the original Item 1, the Secretary ruled that Mr. Miller’s motion was out of order unless it differed materially from the original Item I, and that Mr. Miller’s motion had been accepted on the assumption that it did state a meaning materially different from the original Item I. Mr. Ayres raised a point of order that the motion was out of order because Item VIII had not been taken up immediately after the vote of Item VII. The Secretary stated that the elimination of the exception stated in Item VIII had left only the effective date in that section and that, therefore, he regarded it in the nature of an enacting clause, which if approved would be an approval of the entire report as amended. After voting upon each of the individual items he thought it proper to amend the report as a whole before voting on the final adoption of the re- 152 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

port. If he was wrong in this interpretation, he was subject to being corrected by the house. Mr. Moore oﬀered objections to the use of the words “assigned duties.” The Secretary stated that for the motion to be in order the words must mean that a person could not be assigned by his department, the dean, or the Administration more than one- half of his duties while on full-time employment, but suggested that a clarifying amendment would be in order. Mr. Moore then moved to amend Mr. Miller’s motion to read “no person on modiﬁed service shall be required to carry more than one-half of his duties while on full-time employment.” The motion was seconded and adopted, and the motion thus amended was then adopted.13

Obviously, at least to Moore and others of the “third floor” faculty, efforts were being made to reduce their influence, even to the point of forcing them to reduce the number of classes they taught each semester, thereby radically altering the system which they had developed. Attempts to reduce the influence of the “third floor” mathematics faculty came in a variety of ways. Moore and others in that group had to be constantly alert. Shortly after the debate on the meaning of “modified service,” which had been formally convened by some eleven protests (five from the “third floor” faculty) an effort was made to adopt the motion:

That the number of voting members of the General Faculty who must protest major legislation by the Faculty Council in order to bring it before the General Faculty be increased from ten to twenty.14

It was clear to members of the “third floor” faculty that such a move would reduce their opportunity to argue their academic positions, on such issues as the definitions of “modified service,” before the General Faculty. The rationale presented by the proponents of the motion was that the faculty size had radically increased so it was reasonable to increase the number of protests required. Moreover the number on the Faculty Council had been increased, making it a more representative body. Moore retorted, on the floor of the General Faculty, that as far as he had succeeded in discovering from the Faculty Minutes that he had in his office, there had

13Minutes of the General Faculty, University of Texas, Austin, Texas, March 23, 1954, pp. 6300-6301. 14Minutes of the General Faculty, University of Texas, Austin, Texas, April 1954, pp. 6446-6447. 153 been only six instances in the last five years in which an action of the Faculty Council had been brought before the General Faculty by means of faculty protests against its adoption by the circularization procedure. In five of these six instances the action in question was materially changed. He considered each of these five cases:

1. On March 20, 1950, the Faculty Council adopted a report of the standing committee on admissions. The number of protests against confirmation of this report by the General Faculty was referred to as “more than ten.” At a meeting of the General Faculty on May 9, 1950, a motion to adopt Item of this recommendation was defeated by a vote of 152 to 104. 2. At a meeting of the Faculty Council on May 15, 1950, Pro- fessor Begeman proposed a certain amendment to the up or out” rule. The Faculty Council substituted, for the Bege- man amendment, an amendment proposed by Dr. Dolley. This Dolley amendment, as adopted by the Faculty Coun- cil, was submitted to the General Faculty for adoption by the circularization procedure. There were 11 protests. At the June, 1950, meeting of the General Faculty, Professor Begeman’s amendment was adopted by a vote of 114 to 61. Thus the Faculty Council’s action in substituting the Dol- ley amendment for the Begeman amendment was reversed by a vote of 114 to 61 and there had been only 11 protests against the Faculty Council’s action. 3. There were “more than ten” protests against adoption by the circularization procedure of the Council recommendation concerning the accreditation of ROTC work and its substitution in degree programs. At a General Faculty meeting, Professor Graham offered an amendment which was adopted. 4. There were 16 protests against adoption by the circularization procedure of the Faculty Council recommendation requiring attendance at official commencement activities. “Do I need to remind you of what the General Faculty did about that¿‘ asked Mr. Moore. 5. There were 11 protests against adoption of the Faculty Council recommendation concerning modified service. That also is a matter of recent history. 154 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

Mr. Moore then asked if this record did not have something to do with the real reason why the Faculty Council proposed the change in the number of protests required. He doubted that the increase in the number of elected members on the Faculty Council would suddenly transform the Council into a body none of whose actions would ever need to be reviewed by the Faculty Council. Mr. Moore discounted the idea that if 20 protests were required, they could be obtained. He pointed to the fact that protests were normally not obtained by someone writing one and carrying it around for people to sign. He did not want to do that and thought no one else should have to. He said that all 11 protests against the Council’s recommendations concerning modiﬁed service had been signed only by individuals. Addendum

Mr. Moore has asked that the following statement be added to the Minutes. Mr. Moore has subsequently examined the bound volumes of the Minutes of the General Faculty and Fac- ulty Council and has found that there were 11 protests against the action of the Council on the ROTC case and 20 protests against the adoption of the report of the Standing Committee on Admissions. In the latter case there was one lengthy protest bearing 12 signatures, another signed by 5, and another apparently made by one person and concurred in by another. Only one protest was signed by a single individual. Mr. Moore asked whether ll separate protests might not ordinarily be considered as carrying more weight than 20 signatures secured in the manner indicated. Mr. Williams supported the recommendation of the Faculty Council. He pointed out that action within the Faculty Council was not always unanimous; and if one was looking for protes- tants and needed more than 10, he could in all probability get them in the Faculty Council. Mr. Ettlinger said that precisely because of the reasons advanced for the approval of the recommendation, he was voting against it. The fact that the General Faculty had grown so large was ,he very reason that every facet of the democratic process should be preserved. This Faculty must not abdicate its academic responsibility for fundamental University policy. He stated that history showed that the increasing numbers were required for making group decisions. He quoted from Genesis, Chapter XIX, verses 26-32: 155

If I find fifty righteous men within the city, I will spare all the place for their sakes, and then the numbers are lowered to forty-five, then forty, thirty, twenty and finally the declaration is made if ten be found, then I will not destroy it for the ten’s sake. Who knows but that this city may escape destruction for ten’s sake.

He said although faculty members had the privilege of attending meetings of the Faculty Council, this privilege was of little use because no advance notice was given to the General Faculty as to the business to be transacted. He felt that the General Faculty should not abdicate its responsibility for ultimate policy. The President put the motion to the Faculty, and it was defeated by a vote of 160 to 92.15

Following the defeat of the motion to require more than ten protests to bring legislation proposed by the Faculty Council before the General Faculty, further direct efforts to reduce Moore’s influence and opportunity to teach as he wished were temporarily stymied. In the presence of such remarkable success as that offered by R. L. Moore, what sort of reasons could be good enough to justify placing restrictions on Moore’s teaching? Some of the arguments, on each side, would be used in universities today. Clearly, Moore’s age was becoming a disadvan- tage to him. It could be suggested, or stated, that his mathematics was old and outdated and his own age would only tend to support such a claim might well be true. What of the problem of hiring new faculty? Who would wish to come to Texas and compete with R. L. Moore for students? Might it not be true that the better student would gravitate toward Moore and away from other faculty, or if not true, it might well be thought to be true. Thus some faculty would choose to reject an offer from Texas, perhaps even stating that Moore’s presence would keep them from coming. To an administrator who was trying to hire new faculty, it surely would occur to him that Moore’s presence was an obstacle. There would remain all sorts of points of conflict: How many teaching assistants would each segment of the Department of Mathematics assign? Will there be general comprehensive examinations for doctoral students, departmentally given? Will there be a syllabus for any or each course? As the “third floor” faculty either could not hire new faculty, or chose not to hire new faculty, diverting their resources instead to graduate student support, the preponderance of the number of faculty would swing to

15Minutes of the General Faculty, University of Texas, Austin, Texas, April 1954, p. 6448. 156 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE” the “second floor” segment. This would lead to further efforts to place restrictions on Moore and others of the “third floor.“ Efforts on the part of “third floor” faculty to express their worth were sometimes offered. As those of the “third floor” faculty grew older, they found time and again that they had to argue their case before the General Faculty. Even though the earlier efforts by some to redefine the meaning of modified service (and thereby restrict Moore’s teaching and influence) had been defeated, similar efforts continued to be made. For instance, there later was introduced a motion to disqualify faculty members on modified service from serving on the faculty council and elected faculty committees. That motion read:

A member of the Faculty after reaching modiﬁed service shall not be elected to serve on the Faculty Council or on any committee of the Faculty to which members are elected by the Faculty.16

In discussion of the motion, Ettlinger read the following statement, offering a touch of sarcasm in his play on “modiﬁed service:”

The action of the Faculty Council in voting affirmatively on the report of Professor C. E. Ayres is in the direction of upgrading the Faculty Council and therewith the University of Texas. The stamp of approval by the General Faculty would be a long step in the much desired objective of making the University of Texas a higher educational institution of first class. A member of this faculty who is on “mortified” service has outlived his usefulness and will only stand in the way of a younger member of the faculty. May I remind this faculty that “mortified service” does not necessarily imply diminished usefulness to the University, State and our students. “Mortified” service does mean a half salary, but I know more than one case where it has resulted in continued full responsibility to student, class and scholarly activity. May I take this occasion to detail to this University the outstanding achievement of a colleague of mine. Many exstudents and students are quite excited by the number three standing nationally of our great University in football. How many of our

16Minutes of the General Faculty, University of Texas, Austin, Texas, October, 1961, p. 7778. 157

faculty and administration know that because of Robert Lee Moore’s teaching and work in mathematics the University of Texas is known everywhere in this world as one of the great mathematical centers to be found anywhere. This is not a matter of the past, but it is part of a continuing operation now going on and we hope will continue in the years to come, thanks to Dr. R. L. Moore and another colleague, Dr. H. S. Wall. The last report of the Mathematics Division of the National Research Council was translated into a percentage rating by another colleague, Dr. Fomer V. Craig, and discloses a number one ranking in mathematics for the University of Texas with a percentage of .27, Princeton with .20 was second, Columbia with .18 was third and Harvard with .17 was fourth. The University of Illinois was twentieth with .05. I cite these facts because 1) they should be widely known on this campus and in Texas at least, 2) they point up the kind of reward this report showers on those who are still rendering incomparable service to learning and education. I ask you to vote against this action of the Faculty Council so that this General Faculty may be able to proﬁt from the leadership of men whose scholarly ideals and whose voice is still able and willing to give wise counsel. Vote no on this report.17

Ettlinger’s plea did not carry the day and the motion was passed. How- ever, Ettlinger did support his statements about the strength of the mathematics oﬀerings by ﬁling the statement below in the minutes of the General Faculty.

On January 1, 1955, the National Science Foundation authorized a Survey of Research Potential and Training in the Math- ematical Sciences. This was proposed by the Mathematics Di- vision of the National Research Council. (I was a member of this group at that time.) The Survey Report was issued in two parts on March 15, 1957 and on June 15, 1957. On pp. 83-85 of Part I, are found the publication grouping of the Ph.D. holders from Table I, Ma- jor State Schools; Table II, Major Non-State Schools; Table III, Other State Schools and Table IV, Other Large Non-State

17Minutes of the General Faculty, University of Texas, Austin, Texas, October, 1961, p. 7797. 158 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

Schools. For the purpose of this report only I and II are signif- icant. The number of Ph.D.’s granted by each institution was listed and these holders were divided by the number of publications into four categories. The highest category was labeled P4 and represented the top quality Ph.D. graduates as judged by the amount of published mathematical research. My colleague, Dr. H. V. Craig, made note of this information and constructed a table of ratings from them in the form of a ratio of the number in the fourth and top quality category for each institution to the number of Ph.D. degrees granted by that institution. The top four rankings are as follows: l. University of Texas, .27; 2. Princeton University, .20; 3. Columbia Univer- sity. .18; 4. Harvard University, .17. This gives the University of Texas ﬁrst rank in the production of top quality research mathematicians. This represents for Texas a count of 11 in a total of 41. The number 11 is larger than that of any other in the list of state schools. Among all the non-state schools, the following ﬁve present the data listed:

Total No. of Ph.D.’s Granted Count in Highest Quality Group Brown 108 15 Chicago 296 24 Columbia 94 17 Harvard 186 32 Princeton 177 36

The absolute count for Texas is greater than that of any other University except the five listed above. For these five institutions the ratio for Texas is much higher since Chicago, for example, had only 24 highest quality Ph.D. graduates among a total of 296 degrees granted and similarly in the other cases. Furthermore, all of these five universities except Brown had Ph.D. programs well under way and were producing Ph.D. graduates many years before 1915. Whereas, the University of Texas began (except for one case) in 1923, eight years after the beginning of the Survey. This was a handicap to Texas, and makes the record all the more remarkable, since those who received their Ph.D. degrees from the above five universities during the 159

interim had a larger opportunity to compile a suﬃcient research publication record to achieve a place in the P4 column. Con- sequently, the performance of Texas Ph.D.’s is all the more startling, having been accomplished in many fewer years than the ﬁve universities above except possibly Brown. The other eighteen institutions in Table I and II above, have ratios which go down to .05. Not only are Professor Craig’s ratios implicit in the Survey Re- port but the following quote from Part II, p. 50, last paragraph, has important bearing on the above ratios and is explicitly stated in terms of production of academic mathematicians:

The question whether the graduates of some universities are particularly identiﬁed with academic or non- academic pursuits seems, on the whole, to be answered in the negative. The data of the Survey (Chap- ter IV, Part I, p. 111) are consistent with the hypothesis that almost all the schools are homogeneous in the production of academic and non-academic mathematicians. Probably UCLA, Texas, and Chicago are exceptional, and Iowa State may be. The most applied school in the country seems to be not NYU, or Brown, or MIT, but rather UCLA – evidence, it is sus- pected, of the lure of the aircraft industry. The least applied school seems not to be Harvard or Princeton or Yale, all of which perform in an average manner, but rather Texas.18

Though it often seemed the members of the “third floor” faculty rose to each other’s defense suggesting unusual accord, it remained true that disagreements between them occurred. At one point, questions concerning desegregation arose in discussion on the floor of the faculty. Ettlinger read a statement in support of according “all students of the University equal status in all respects in housing and eating facilities and in student activities.” He argued that this would be in the direction of a “first class” university. Moore then asked Ettlinger if he considered Princeton and Harvard of the first class? Did they discriminate against anyone? When Ettlinger answered that he didn’t know, Moore pointed out that they discriminated against women. As the discussion continued, Moore “asked for the name of

18Minutes of the General Faculty, University of Texas, Austin, Texas, November 1961, p. 7803. 160 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE” one person of first class ability who had refused to come to the University because of segregation.” Two faculty indicated that they had considered that aspect of the University with grave reservation before coming to Texas. The Minutes record that Moore stated “he believed that integration would have nothing to do with recruiting first class men for the University.” Stu- dent lore, gained from reports of those at the meeting, is to the effect that Moore said, “I asked for a first rate man who did not come here, not for two second rate men who did come here.” The attempts from without and within to alter the activities of the Department of Pure Mathematics, and then the “third floor” of the De- partment of Mathematics, gave rise to a general state of upset among the students and faculty alike. The friction “between floors” grew to the point that students often felt more comfortable if they pursued one of the paths, and not mix courses from both factions. Then too, the ages of the third floor faculty, particularly Moore and Ettlinger, caused some graduate students to seriously question whether they should transfer elsewhere. Yearly, it seemed more likely that a student beginning graduate study might not be able to finish with Moore, Ettlinger, or perhaps any of the other of the “third floor” group. In the presence of such difficulties, it is interesting to take note of Moore’s contributions. Almost half (24 of 50) of his doctoral students graduated after he began “modified service.” His publications ceased after 1953, except for a major revision of his colloquium publication, which appeared in revised form in 1962, some thirty years after the first publication. There are those who hold fast to the opinion that Moore intentionally ceased his publishing of new results, instead leaving those questions to his students for dissertation purposes. Others believe he was too busy to engage in research at a level comparable to earlier years. His physical stamina was tested as he continued teaching five courses each semester, sometimes splitting his advanced group into several sections so as to handle them better. The surplus of students of the post war years had allowed the formulation of an extensive array of graduate students who were hired as teaching assistants. There were many students seeking his direction; he was teaching as many courses as he ever had and more than the normal teaching load of most of the graduate faculty; he was distracted at the several efforts to pass rules or regulations which would have the effect of restricting his teaching; some arrangements were becoming more difficult within the department as renewed efforts were being made to introduce a syllabus approach into courses at each level, including departmental qualifying examinations for doctoral students. His influence on students was strong, not only as regards their mathematics. It is not uncommon that a major professor exerts exceptional 161 influence on his students; Moore accomplished it in rare style. Though his students “thought for themselves,” his manner of exact speech was taken up by many of his students. His influence on some went very far. At one point it is reported that Moore formulated a theory that successful research mathematicians suffered from hay fever. Resulting from that theory was the experience of a student who would lie for hours under cedar bushes outside Austin, until he too developed a reaction that could be identified as hay fever.19 Moore handled his classes in such a fashion that no doubt could arise as to the manner or integrity of the proceedings. No mention would be made of behavior of the students regarding written examinations. A student might well spend much time on a written examination, which might include one or more unsolved problems, and turn it in by sliding it under the door of Moore’s office, long after Moore had departed. While no official declaration of an honor system was made, Moore’s practice exceeded in honor that which is often obtained by spelling out the formalities of an honor system. The obligation of the student was real in many ways. As he became more advanced, he was expected to develop his skills in determining whether he had actually settled a question, as well as developing the mental power to settle it. A student who clarified a result was expected to have worked cut the details, as well as having seen the general approach leading to solution. At one point, an advanced student who had often claimed a result, only to expose the general argument but with the defect of some unexpected detail surprise him, claimed that he had settled a question raised by Moore. Moore told the student to show his argument to another more advanced student, and if it be correct with every detail, he would present it in class. The student before long indicated the argument to another graduate student who was unaware of Moore’s instructions. The proof was generally correct, though not all details were presented, nor asked for by the second graduate student. The “refereeing” graduate student was shortly surprised to be confronted by Moore and the student who claimed the argument. Moore inquired, “Did Mr. show you a proof of ?” The answer given was “Yes.” Next came, “Did he show you every detail?” The answer was “no,” though it surely did seem to the refereeing student that any detail which had been omitted was not of serious question, so had not asked to see it. At that point, Moore terminated the discussion and at his next class with the student, called upon another member of the class who also claimed the result. The argument presented was correct and formed a basis for a dissertation. The student who had not offered each detail never completed

19Personal conversation with R. L. Wilder, Santa Barbara, California, December 4, 1971. 162 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE” his dissertation. Moore was sometimes thought to be severe with such discipline. At times he would punish a class, or individuals within it, by stating major theorems without intermediate lemmas. Extreme discouragement would set in. Some members of the class would drop out. Others would be almost to that point before they would “catch fire” and go on to successfully complete their doctoral work under Moore. No doubt, Moore observed much talent drop away from his courses as he chose to challenge in a particular fashion one or more students in a class. It amounted to a matter of judgement on his part as to how to proceed with a certain class or certain students in that class. His success testifies to his judgement. As one administrator stated, “ . . . I just assumed he had an extra trigger in his head that did it.”20 Most of the unusually mature graduate students, caused by World War II, had moved off the scene by the time Moore began his years of modified service. Few of his twenty-four graduates after 1953 saw military service in World War II. Though many have attempted to do so, comparison of the before 1953 graduates with those of the post 1953 period is difficult, perhaps for some of the same reasons that comparisons between Jack Dempsey and Rocky Marciano are difficult. Those who argued for Moore’s retirement after 1953 were wont to compare the two groups. Today, in terms of national reputation, recognition, or high office held in professional societies, the lustre of the pre-1953 group is greater than that of the post 1953 group. However, there exists unusual strength among the post-1953 group and it is yet early to declare the abler of the two groups. Moore’s mathematical interests can be identified by examining the topics of his students” dissertation work. His first three students after 1953 dealt with spirals in the plane. Another dealt with local separability properties, one with dense metric subspaces of a space, several dealt with various questions related to continua and others dealt with upper-semi-continuous decompositions. Curiously, Moore’s publications (except for revision of his colloquium book) terminated with his going on modified service. Just a few days past his seventieth birthday, he communicated a paper, “Spirals in the plane” to the Proceedings of the National Academy of Sciences. His first three students, following his going on modified service, dealt with the topic of Moore’s last paper in their own dissertation. Though not many years have passed since those students of the post- 1953 years graduated, some evaluation of their worth as academic mathematicians is possible by comparing their contributions with results on national surveys. In 1967 there appeared in the American Mathematical

20Personal conversation with Norman Hackerman, Houston, Texas, December 11, 1970. 163

Monthly (Vol. 74, pp. 1126-1129) the results of a national survey which gives the average numbers of publications of doctorates in mathematics who graduated in 1950-1959 from various universities. The three highest aver- ages were 6.3 publications per doctorate from Tulane, 5.44 from Harvard, and 4.96 from the University of Chicago. For the same period, considering on y those papers reviewed in The Mathematical Reviews, Moore’s students averaged 7.1 and, by the fall of 1970, averaged 9.5. Perhaps of even more interest is the fact that, considering only those papers which are reviewed in The Mathematical Reviews, his students who graduated in 1960-1964 averaged eight publications per student by the fall of 1970. Recognition of Moore’s extraordinary feat of producing such eﬀective students came in many forms. An article appeared on December 10, 1967 in the Dallas Morning News, stating

Because of Dr. Moore’s former students, the University of Texas is the only state university with more than one mathematics doctoral graduate listed in the National Academy of Sciences. Only two others, the University of Illinois and the State Univer- sity of Iowa, have any graduates within this distinguished circle. The latter schools have one NAS member each.

That Moore was, in many respects, a teacher extra ordinary is indicated by a report, issued in 1961 by a presidentially appointed commission. Entitled “The Central Purpose of American Education,” it deﬁned that purpose as being the development of a student’s power at rational thought. The following paragraph supported that deﬁnition:

The rational powers of any person are developed gradually and continuously as and when he uses them successfully. There is no evidence that they can be developed in any other way. They do not emerge quickly or without effort. The learner of any age, therefore, must have the desire to develop his ability to think. Motivation of this sort rests on feelings of personal adequacy and is reinforced by successful experience. Thus the learner must be encouraged in his early efforts to grapple with problems that engage his rational abilities at their current level of development, and he must experience success in these efforts.21

By 1961 Robert Lee Moore had been practicing that sort of activity in his teaching for at least ﬁfty years.

21The Central Purpose of American Education. The Educational Policies Commission of the National Education Association of the United States, 1961. 164 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

The Furling of the Flag

Empires rise and fall. The glory that was Rome’s can not be recaptured. Socrates lived and taught. R. L. Moore came to Texas and taught. It may be again as many years as those separating Socrates and Moore before the likes of either is seen again. In many respects, Moore was at his teaching strength toward the end of the 1960’s. Simultaneously, efforts to cause his retirement seemed to be peaking. To say the situation was complex is not to say there was no right nor wrong of it. Universities of today would do well to hear the values offered by those at Texas who grappled with the question of Moore’s retirement. Few universities ever experience such an opportunity to so well expose their academic values. Examination of the factors surrounding Moore in the late 1960’s almost requires examination of the populations which were reasonably well defined: Students, both undergraduate and graduate, “second floor” and ” third floor;” faculty, both “second floor” and “third floor,” those who were candidates for positions at Texas, and the graduates of Texas; and administrators. Any such separation introduces oversimplification but perhaps allows expression of some of the moods and rationales of those who were active in the decision. Undergraduate students, by the late 1960’s, were for the most part enrolled in “second floor” courses. In no sense was it true that “second floor” faculty considered themselves unanimously applied. Indeed, many of the faculty felt not content that courses they taught carried the designation of “E” behind the number of the course, suggesting some sort of applied offering. Undergraduate students who wished to take courses from Moore almost had to begin with him. Otherwise, faculty not in sympathy with Moore might too early tell them “too much,” thereby disqualifying them for classes with Moore. Transfer students, particularly advanced ones often would not be admitted to Moore’s classes since they often would have been exposed to concepts elsewhere that Moore would introduce in his own fashion. So there would be unhappiness on the part of some students who would be turned away from his classes. Others, gaining admittance to his class, would not like what they found. His style of teaching would be foreign to most they had experienced: no books; no collaboration; no subject lectures; an uncompromising demand for their talent, if any. Protests would result. Stories had abounded for years between students about the mathematics, which had come to be known as “third floor” mathematics and the faculty which had become known as “third floor” faculty, until stories had become legend. Such stories inhibited some students from testing the flavor of the third floor and, simultaneously, quickened the interest of another, smaller 165 group. The graduate students, while hardly better defined, were a smaller group than the undergraduates. While Wall (1946) and Lane (1949) were the only two to join the “third floor” group after the war, the “second floor” faculty had increased at a more rapid rate. Competition for graduate student support, many as teaching assistants caused uncertainty among graduate students, influencing under whom they might try to work, and caused difficulty between faculty as distribution was made between “second” and “third” floor groups. Often “third” floor graduate students would be assigned to teach “E” or “second” floor courses. Some of such students, being inflamed at the method of teaching encountered on the “third floor,” applied it to their own classes. In some instances it might be done quite successfully and in other instances would amount to a poor decision. In almost every instance though, student discontent would arise, causing disagreements between all who became involved. The friction evident between faculty manifested itself among the graduate students, separating them into two rather well defined groups: those who were working under third floor faculty and those who were not. Faculty were split into two groups, identified as “third floor” and those not among the “third floor” faculty. By the late 1960’s the third floor group consisted of Moore, Wall, Lubben, and Ettlinger. Only Wall was young enough to sit on the departmental Budget Council. Moore and Ettlinger were well past seventy years of age and Lubben was nearing that age. As candidates for faculty positions were proposed, often the third floor group protested, feeling enough strength was already about, recalling the success experienced by German universities which had few faculties.

But you look at places like G¨ottengin- back in their great days. They didn’t have a whole raft of faculty. Most of their teaching was done by those Assistants, or what ever they call them. I suppose that’s about like Teaching Assistants. If you can get Teaching Assistants that can teach all the undergraduate work, they do a better job than some of these others.22

That same hesitancy on the part of Moore, Wall, Lubben, and Ettlinger was not well received by others. As enrollments skyrocketed and federal funding became more plentiful, it became a national phenomena to hire as many faculty as possible, and to offer reduced teaching loads to faculty to be competitive, encourage research, and reward research. Thus there were at least two philosophies at work; one offered by Moore and others of the “third floor” to continue doing what they did so well; the other calling

22Personal conversation with H. S. Wall, Austin, Texas, February 13, 1971. 166 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE” for change toward the pattern emerging nationally elsewhere, hiring more faculty, gaining research funding, and generally being of the ilk of those departments generally held in high esteem. By the mid-1960’s the faculty of the “third floor” had been reduced to four: Moore, Lubben, Ettlinger, and Wall. Their students, who were hired as teaching assistants, were generally officed there, or on the fourth floor of Benedict Hall. As new faculty were added, competition arose for the third floor officing among the faculty, giving rise to another point of friction. Toward the end of the 1960’s Dean Burdine, of the College of Arts and Sciences, died and was followed in office by J. R. Silber, a member of the Philosophy Department, who had gained reputation for success as a chairman of that department. He accepted his new post with commitment toward gaining national reputations for departments in his college. Visiting scholars were brought onto the campus to assist the new dean in his efforts to evaluate current departmental strengths, potential for developing further strength, and procedures to take to gain that desired strength. In many ways, he heard criticisms of the mathematics department as it was when he became dean. Some of those criticisms included: The department, though formally one, is really functioning as if split, • The faculty is too few in number for the number of students to be • taught, Entirely too many graduate students, as teaching assistants, teach • too many advanced courses, Teaching loads need be reduced to attract quality faculty, (5) The • informal but deep split in the department, and Moore’s presence, hindered the hiring of new faculty. These, and other critical statements, were heard by Silber both from within and without the department and had been heard for a number of years by those holding administrative posts at Texas. Silber determined to take the steps which he felt others had wanted to take but did not. Indeed, an attempt to make mandatory the retirement of all faculty at age seventy-five had gained the approval of the Board of Regents in 1967. In part, that ruling read: Beginning September 1, 1967, members of institutional faculties and institutional non-teaching staffs must retire completely at the end of the fiscal year that includes their seventy-fifth birthday.23 23Minutes of the Board of Regents, University of Texas, Austin, Texas, Meeting 652, 1967, p. 109. 167

That single ruling stimulated much activity among students at Texas and faculties at other universities. The members of the Board of Regents we-re bombarded with letters, telephone calls, and telegrams, asking that the matter be reconsidered, pointing out that the loss of Moore from the faculty would suﬀer deep injury to the University of Texas. Resulting on January 26, 1968, was a rescinding of the earlier approved mandatory retirement age of seventy-ﬁve, stating in part:

. . . striking at the end of each of these sections the words “except that, beginning September 1, 1967, no member will be continued on modified service beyond the fiscal year in which his seventy-fifth birthday occurs.” 24

Retirement for Moore thus was prevented again. It was seen by some as a victory for academic propriety and by others as one more delay before the Department of Mathematics could be about its business of becoming a modern, effective department. Any representation of the events following 1967 would be incomplete and would not accurately portray the contributions of the many people involved. Indeed, it does appear impossible to gain an evaluation of those proceedings so well that the influence of each person and each event might be effectively waged. One source though, does illustrate the agony confronting the “third floor” students, whose progress toward a degree was blunted or aborted because of the turmoil rampant during 1967 and 1968 around Moore. Students at the University of Texas, who wished to continue studying with Moore, recalled Socrates, whose birth was about 469 B. C. Some seventy years later Socrates accepted a lethal dose of hemlock, with the decision that such punishment be administered to him made by men of less clear minds. Some Texas students felt they were seeing a modern-day re-enactment of the tragedy of Socrates. The hemlock that was to be administered to Moore would come in the form of retirement. In an age of student revolt, the Texas students entered into an activity of “constructive rebellion” attempting to prevent their tragedy by working within the system. They cajoled, demanded, made formal oral presentations and showed marked resourcefulness as they attempted to preserve a rare teacher for their own benefit. They became an embattled group as they strove to retain the “third floor” philosophy and faculty. Their main weapon was that of the glorious past contributions and effectiveness of that group. It was not unlike Robert E. Lee’s Army of Northern Virginia whose past valor and deeds made adversaries timid, even during the retreat to Appomattox.

24Minutes of the Board of Regents, University of Texas, Austin, Texas, Meeting 658, January 26, 1968, p. 16. 168 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE”

Students, fearing that Moore would be retired, transferred to other universities. Faculty were being hired with the understanding that Moore would be retired. Even those who were ﬁrm disciples of Moore might hesitate to take a position at Texas. One wondered about himself as to his feelings about being on the same faculty with Moore:

Now, what should I do if a student were to come to me and ask to study with me? Should I say, “Come ahead,” or should I say, “You really should go down the hallway and take a course with Professor Moore. You might not see his teaching talent for another three thousand years . “ 25

Departmental examinations, required of all doctoral students, were being initiated. To students of Professor Moore, and others of the old Pure Mathematics faculty, it seemed unnatural that a man would learn certain subject matter and then suddenly be able to perform research some day. Instead their experience was that they learned, from the earliest levels to do it for and by themselves. Each day they were rigorously examined in their class. A former student recalls:

. . . every class meeting was, in fact, an oral examination. Every class meeting of every graduate course, one was examined, not formally, perhaps, but, if after a few weeks in a graduate course, you didn’t have anything to contribute, why then, it was just understood that you just weren’t doing your job. And so, actually, though nothing was spelled out in the way of formal examinations, the examining process was extremely severe. It went on every time a class met!26

The new faculty which were added were, for the most part, of a background and training different from that espoused by Moore and others of the old Pure Mathematics faculty. Efforts to revise the curriculum and modernize the program were underway. Reduced teaching loads and “mon- ster” sections were being instituted. Multiple sections of each course gave basis for arguments in support of a syllabus for each course. If a university were to look to the future, could it afford to do so with the values of an eighty-five year old professor? Eventually, in 1969, after three appeals for such, a recommendation came forth from the Department of Mathematics that was heard by the

25Personal conversation with John M. Worrell, Jr., Houston, Texas, January 1971. 26Personal conversation with John S. Mac Nerney, Houston, Texas, June 14, 1971. 169 administration at the University as a recommendation against Moore’s continuance. Students made strong effort at this point to preserve their opportunity to study with Professor Moore. They compiled a massive report, “Concerning Dean John R. Silber and The Proposed Dismissal of Professor R. L. Moore” 27 and finally made a presentation, along with other graduated mathematicians, before the Board of Regents. Their early optimism following that meeting was short lived. It soon thereafter became known that Moore would be retired since it would only further damage the department as a result of the turmoil which would be caused by re-opening the question of his retirement. Thus it was that the last day of the summer session 1969, Robert Lee Moore taught his last class at Texas. The class closed and no student moved, as if they could prevent it from happening by not leaving. Finally, Moore walked from the room. The career of Moore, as a university professor was closed. A few years earlier he had offered a statement about the meaning of a university of the first class:

I do not claim to know how to deﬁne a university of the ﬁrst class, but I wonder whether anyone here would challenge the assertion that no university is of that class unless (1) a very substantial amount of really fundamental research of a high order is carried on by members of its faculty, and (2) there are some members of its faculty who are intensely on the alert to discover and develop outstanding research ability on the part of their students and who are both capable of recognizing such ability in the early stages of its manifestation and of developing it when it is discovered.28

The magniﬁcence of his contributions graced the University of Texas for many years.

27Concerning Dean John R. Silber and The Proposed Dismissal of Professor R. L. Moore, University of Texas Library, Austin, Texas. 28Minutes of the General Faculty, University of Texas, Austin, Texas, May 15, 1945, p. 3151. 170 CHAPTER 6. THE YEARS OF “MODIFIED SERVICE” Part II

171

Chapter 7

The Doctoral Students of Robert Lee Moore.

173 174 CHAPTER 8 University of Pennsylvania Kline, John R., 1916 Hallett, George H., 1918 Mullikin, Anna M., 1922

University of Texas

Wilder, Raymond L., 1923 Pearson, Bennie J., 1955 Lubben, Renke G., 1925 Armentrout, Steve, 1956 Whyburn, Gordon T., 1927 Mahavier, William S., 1957 Roberts, John H., 1929 Treybig, L. Bruce, 1958 Cleveland, Clark M., 1930 Younglove, James N., 1958 Dorroh, Joe L., 1930 Henderson, George W., 1959 Vickery, Charles W., 1932 Worrell, John M., Jr., 1961 Klipple, Edmund C., 1932 Cook, Howard, 1962 Basye, Robert E., 1933 Cornette, James L., 1962 Jones, F. Burton, 1935 Reed, Dennis K., 1965 Swain, Robert L., 1941 Baker, Harvey L., Jr., 1965 Sorgenfrey, Robert H., 1941 Baker, Blanche, 1965 Miller, Harlan C., 1941 Davis, Roy D., Jr., 1966 Young, Gail S., 1942 Rogers, Jack W., Jr., 1966 Bing, R. H., 1945 Secker, Martin D., 1966 Moise, Edwin, E., 1947 Cook, David E., 1967 Anderson, Richard D., 1948 Hinrichsen, John W., 1967 Rudin, Mary E. Estill, 1949 O’Connor, Joel L., 1967 Burgess, Cecil E., 1951 Green, John W., 1968 Ball, Billy Joe, 1952 Proﬃtt, Michael H., 1968 Dyer, Eldon, 1952 Purifoy, Jesse A., 1969 Hamstrom, Mary-Elizabeth, 1952 Jackson, Robert E., 1969 Slye, John M., 1953 Stevenson, Nell Elizabeth, 1969 Mohat, John T., 1955 Chapter 8

Publications of Robert Lee Moore

175 176 CHAPTER 7

1. The betweenness assumptions (written up by G. B. Halsted). Amer- ican Mathematical Monthly 9 (1902) 152. 2. Geometry in which the sum of the angles of every triangle is two right angles. Trans. Amer. Math. Soc., 8 (1907) 369-378. 3. Sets of metrical hypotheses for geometry. Trans. Amer. Math. Soc., (1908) 487-512. Doctoral dissertation. 4. A note concerning Veblen’s axioms for geometry. Trans. Amer. Math. Soc. , 13 (1912) 74-76. 5. On Duhamel’s theorem. Ann. Math. 13 (1912) 161-166. 6. The linear continuum in terms of point and limit. Ann. Math. 16 (1915) 123-133. 7. On the linear continuum. Bull. Amer. Math. Soc. 22 (1915) 117-122. 8. On a set of postulates which suﬃce to deﬁne a number-plane. Trans. Amer. Math. Soc. 16 (1915) 27-32. 9. Concerning a non-metrical pseudo-Archimedean axiom. Bull. Amer. Math. Soc. 22 (1916) 225-236. 10. On the foundations of plane analysis situs. Trans. Amer. Math. Soc. 17 (1916) 131-164; brief summary in NAS Proc. 2 (1916) 270-272. 11. A theorem concerning continuous curves. Bull. Amer. Math. Soc. 23 (1917) 233-236. 12. A characterization of Jordan regions by properties having no reference to their boundaries. Proc. Nat. Acad. Sci. 4 (1918) 364-370. 13. Concerning a set of postulates for plane analysis situs. Trans. Amer. Math. Soc. 20 (1919) 169-178. 14. Continuous sets that have no continuous sets of condensation.Bull. Amer. Math. Soc. 25 (1919) 174-176. 15. On the most general class L of Frechet in which the Heine-BorelLebesgue theorem holds true. Proc. Nat. Acad. Sci. 5 (1919) and Kline, J. R. 16. On the most general plane closed point-set through which it is possible to pass a simple continuous arc. Ann. Math. 20 (1919) 218-223. 17. On the Lie-Riemann-Helmholtz-Hilbert problem of the foundations of geometry. Amer. J. Math. 41 (1919) 299-319. PUBLICATIONS OF ROBERT LEE MOORE 177

18. Concerning simple continuous curves. Trans. Amer. Math. Soc. 21 (192) 333-347.

19. Concerning certain equicontinuous systems of curves. Trans. Amer. Math. Soc. 21 (1921) 41-55.

20. Concerning connectedness im Kleinen and a related property.Fund. Math. 3 (1922) 232-237.

21. Concerning continuous curves in the plane. Math. Zeit. 15 (1922) 254-260.

22. On the relation of a continuous curve to its complementary domains in space of three dimensions. Proc. Nat. Acad. Sci. 8 (1922) 33-38.

23. On the generation of a simple surface by means of a set of equicontinuous curves. Fund. Math. 4 (1923) 106-117.

24. Concerning the cut-points of continuous curves and of other closed and connected point sets. Proc. Nat. Acad. Sci. 9 (1932) 101-106.

25. An uncountable, closed, and non-dense point set each of whose complementary intervals abuts on another one at each of its ends. Bull. Amer. Math. Soc. 29 (1923) 49-50.

26. Report on continuous curves from the viewpoint of analysis situs. Bull. Amer. Math. Soc. 29 (1923) 289-302.

27. Concerning the sum of a countable number of mutually exclusive continua in the plane. Fund. Math. 6 (1924) 189-202.

28. Concerning the common boundary of two domains. Fund. Math. 6 (1924) 203-213.

29. Concerning relatively uniform convergence.Bull. Amer. Math. Soc. 30 (1924) 504-505.

30. An extension of the theorem that no countable point set is perfect. Proc. Nat. Acad. Sci. 10 (1924) 168-170.

31. Concerning the prime parts of certain continua which separate tbe plane. Proc. Nat. Acad. Sci. 10 (1924) 170-175.

32. Concerning upper semi-continuous collections of continua which do not separate a given continuum.Proc. Nat. Acad. Sci. 10 (1924) 356-360. 178 CHAPTER 7

33. Concerning seti of se;eents whlch cover s polnt eet In the Vitali sense. Proc. Net. Aced. Sci. 10 (1924) 464-467. 34. Concerning the prime parts of a continuum. Math. Zeit. 22 (1925) 307-315. 35. A characterization of a continuous curve. Fund. Math. 7 (1925) 36. Concerning the separation of point sets by curves. Proc. Nat. Acad. Sci. 11 (1925) 469-476. 37. Concerning upper semi-continuous collections of continua. Trans. Amer. Math. Soc. 27 (1925) 416-428. 38. Concerning the relation between separability and the proposition that every uncountable point set has a limit point. Fund. Math. 8 (1926) 189-192. 39. Conditions under which one of two given closed linear point sets may be thrown into the other one by a continuous transformation of a plane into itself. Amer. J. Math. 48 (1926) 67-72. 40. Covering theorems. Bull. Amer. Math. Soc. 32 (1926) 275-282. 41. A connected and regular point set which contains no arc. Bull. Amer. Math. Soc. 32 (1926) 331-332. 42. Concerning indecomposable continua and continua which contain no sub sets that separate the plane. Proc. Nat. Acad. Sci. 12 (1926) 43. Concerning paths that do not separate a given continuous curve. Proc. Nat. Acad. Sci. 12 (1926) 745-753. 44. Some separation theorems. Proc. Nat. Acad. Sci. 13 (1927) 711-716. 45. Concerning triods in the plane and the junction points of plane continua. Proc. Nat. Acad. Sci. 14 (1928) 85-88. 46. On the separation of the plane by a continuum. Bull. Amer. Math. Soc. 34 (1928) 303-306. 47. A separation theorem. Fund. Math. 12 (1928) 295-297. 48. Concerning triodic continua in the plane. Fund. Math. 13 (1929) 261-263. 49. Concerning upper semi-continuous collections. Monatsh. Math. Phys. 36 (1929) 81-88. PUBLICATIONS OF ROBERT LEE MOORE 179

50. Foundations of Point Set Theory. American Mathematical Society Colloquium Publication , 13 (1932) vii+486. 51. Concerning compact continua which contain no continuum that separates the plane. Proc. Nat. Acad. Sci. 20 (1934) 41-45. 52. A set of axioms for plane analysis situs. Fund. Math. 25 (1935) 13-28. 53. Fundamental theorems concerning point sets: I, Foundations of a point set theory of spaces in which some points are contiguous to others; II, Upper semi-continuous collections of the second type; III, On the structure of continus. Rice Institute Pamphlets 23 (1936) 1-74. 54. Concerning essential continua of condensation. Trans. Amer. Math. Soc. 42 (1937) 41-52. 55. Review of a book by Veblen and Young in Bull. Amer. Math. Soc. (1920) 56. Concerning accessibility. Proc. Nat. Acad. Sci. 25 (1939) 648-653. 57. Concerning the opne subsets of a plane continuum.Proc. Nat. Acad. Sci. 26 (1940) 24-25. 58. Concerning separability.Proc. Nat. Acad. Sci. 28 (1942) 56-58. 59. Concerning intersecting continua. Proc. Nat. Acad. Sci. 28 (1942) 60. Concerning a continuum and its boundary. Proc. Nat. Acad. Sci. 28 (1942) 550-555. 61. Concerning domains whose boundaries are compact. Proc. Nat. Acad. Sci. 28 (1942) 555-561. 62. Concerning continua which have dendratomic subsets. Proc. Nat. Acad. Sci. 29 (1943) 384-389. 63. Concerning tangents to continua in the plane. Proc. Nat. Acad. Sci. 31 (1945) 67-70. 64. A characterization of a simple plane web. Proc. Nat. Acad. Sci. 32 (1946) 311-316. 65. Spirals in the plane. Proc. Nat. Acad. Sci. 39 (1953) 207-213. 66. Foundations of Point Set Theory. (revision) American Mathematical Society Colloquium Publications 13 (1962). 180 CHAPTER 7 Chapter 9

Academic Descendants of R. L. Moore and Individual Publications

181 182 CHAPTER 9 J. R. Kline and his Mathematical Descendants

01 Kline, J. R. 02 Gehman, H. M , University of Pennsylania, 1925 03 Strebe, David D., University of Buffalo, 1952 03 Berhns, Vernon N., University of Buffalo, 1953 03 Gough, Lillian, University of Buffalo, 1953 03 Montgomery, Mabel, D., University of Buffalo, 1953 03 Warner, Frederick C., University of Buffalo, 1953 02 Flanders, Donald, University of Pennsylvania, 1927 02 Ayres, William L., University of Pennsylvania, 1927 03 Simond, Ruth, University of Michigan, 1938 03 Vance, Elbridge P., University of Michigan, 1939 03 Ollman, Loyal Frank, University of Michigan, 1939 03 Schnechenberger, Edith R., University of Michigan, 1940 04 Feidner, Jean B., University of Buffalo, 1953 04 Belate, Edward J., University of Buffalo, 1958 04 Tidd, Richard F., University of Buffalo, 1959 04 Uschold, Richard L., SUNY at Buffalo, 1963 03 Odle, John W., University of Michigan, 1941 03 Miller, Don D., University of Michigan, 1941 02 Benton, Thomas C., University of Pennsylvania, 1929 02 Zippin, Leo, University of Pennsylvania, 1929 02 Rutt, N. E., University of Pennsylvania, 1929 02 Kusner, J. H., University of Pennsylvania, 1931 02 Smith, Adam 3., University of Pennsylvania, 1934 02 Claytor, Shieffelin, University of Pennsylvania, 1934 02 Milgram. Arthur N., University of Pennsylvania, 1937 03 Exner, Robert, Syracuse University, 1949 03 Kostenbauder, Adnah, Syracuse University, 1952 02 Hemmingsen, Erik, University of Pennsylvania, 1946 03 Peterson, Bruce B., Syracuse University, 1962 03 Reddy, William L., Syracuse University, 1964 03 Abramson, Paul, Syracuse University, 1965 03 Antonelli, Peter, Syracuse University, 1966 03 Blackwell, Paul, Syracuse University, 1968 02 Barrett, Lida K., University of Pennsylvania, 1954 03 Wiginton, C. Lamar, University of Tennessee, 1964 04 Barr, Betty, University of Houston, 1971 ACADEMIC DESCENDANTS AND PUBLICATIONS 183

04 Engvall, John, University of Houston, 1972

Publications of J. R. Kline and his Mathematical Descendants

Kline J. R. 01 • 1. Concerning the complement of a countable infinity of point sets of a certain type. Bull. Amer. Math. Soc. 23 (1917) 290-292. 2. The converse of the theorem concerning the division of a plane by an open curve. Trans. Amer. Math. Soc. 18 (1917) 177-184. 3. A definition of sense on closed curves in non-metrical analysis situs. Ann. of Math. 19 (1919) 185-200. 4. Concerning sense on closed curves in non-metrical plane analysis situs. Ann. of Math. 21 (1918) 113-119. 5. On the passing of simple continuous arcs through plane point sets. Tohoku Math. J. 18 (1920) 116-125. 6. A new proof of a theorem due to Schoenflies. Proc. Nat. Acad. Sci.6 (1920) 529-531. 7. Concerning approachability of simple closed and open curves. Trans. Amer. Math. Soc. 21 (1920) 451-458. 8. A theorem concerning connected point sets. Fund. Math. 3 (1922) 238-239. 9. Closed connected point sets which are disconnected by the removal of a finite number of points. Proc. Nat. Acad. Sci. 9 (1923) 7-12. 10. Closed connected sets which remain connected upon the removal of certain subsets. Fund. Math. 5 (1924) 3-10. 11. Concerning the division of the plane by continua. Proc. Nat. Acad. Sci. 10 (1924) 176-177. 12. Concerning the complementary intervals of countable closed sets. Bull. Amer. Math. Soc. 31 (1925) 409-410. 13. Concerning the sum of two continua each irreducible between the same pair of points. Fund. Math. 7 (1925) 31-322. 14. A condition that every subcontinua of a continuous curve be a continuous curve. Fund. Math. 10 (1927) 298-301. 184 CHAPTER 9

15. Separation theorems and their relations to recent developments in analysis situs. Bull. Amer. Math. Soc. 34 (1928) 155-192. 16. What is the Jordan curve theorem? Amer. Math. Monthly 49 (1942) 281-286 . 17. On the most general plane closed point set through which it is possible to pass a simple continuous ARC. Ann. Math. 20 (with Moore, R. L.) (1919) 218-233.

Gehman, H. M. 02 • 1. Concerning the subsets of a plane continuous curve. Ann. of Math. 2 (1925) 29-46. 2. On irredundant sets of postulates. Bull. Amer. Math. Soc. 32 (1926) 159-161. 3. On extending a continuous correspondence of two plane continuous curves to a correspondence of their planes. Trans. Amer. Math. Soc. 28 (1926) 252-265. 4. Some conditions under which a continuum is a continuous curve. Ann. of Math. 27 (1926) 381-384. 5. Concerning irreducible connected sets and irreducible continua. Proc. Nat. Acad. Sci. 12 (1926) 544-547. 6. Some relations between a continuous curve and its subsets. An- nals. of Math. 28 (1927) 103-111. 7. Irreducible continuous curves. Amer. J. Math. 49 (1927) 189196 . 8. Concerning acyclic continuous curves. Trans. Amer. Math. Soc. 29 (1927) 553-568. 9. Concerning end points of continuous curves and other continua. ibid. 30 (1928) 53-84. 10. Concerning certain types of non-cut points, with an application to continuous curves. Proc. Nat. Acad. Sci. 14 (1928) 431-433. 11. Concerning irreducible continua. ibid., 433-435. 12. On extending a continuous (1,1) correspondence (second paper). Trans. Amer. Math. Soc. 31 (1929) 241-252. 13. On extending a correspondence in the sense of Antoine. Amer. J. Math. 51 (1929) 385-396. 14. Centers of symmetry in analysis situs. Amer. J. Math. 52 (1930) 543-547. ACADEMIC DESCENDANTS AND PUBLICATIONS 185

15. A special type of upper semi-continuous collection. Proc. Nat. Acad. Sci. 16 (1930) 609-613. 16. Concerning sequences of homeomorphisms. ibid. 18 (1932) 460- 465. 17. On extending a homeomorphism between two subsets of spheres. Bull. Amer. Math. Soc. 42 (1936) 79-81.

Strebe, David D. 03 • 1. Irreducibly connected spaces. Duke Math. J. 20 (1953) 551-561. 2. Irreducible closed connexes. Duke Math. J. 22 (1955) 365-372.

Ayres, W. L. 02 • 1. A new characterization of plane continuous curves. Bull. Amer. Math. Soc. 33 (1927) 201-208. 2. Concerning continuous curves and correspondences. Ann. of Math. 28 (1927) 396-418. 3. Concerning the boundaries of domains of continuous curves. Bull. Amer. Math. Soc. 33 (1927) 565-571. 4. Note on a theorem concerning continuous curves. Ann. of Math. 28 (1927) 501-502. 5. On the structure of a plane continuous curve. Proc. Nat. Acad. Sci. 13 (1927) 749-754. 6. On the separation of points of a continuous curve by arcs and simply closed curves. ibid. 14 (1928) 201-206. 7. An elementary property of bounded domains. Bull. Amer. Math. Soc. 34 (1928) 200-204. 8. Concerning the arc-curves and basic sets of a continuous curve. Trans. Amer. Math. Soc. 30 (1928) 567-578. 9. Concerning continuous curves of certain types. Fund. Math. 11 (1928) 132-140. 10. On continuous curves in n dimensions. Bull. Amer. Math. Soc. 34 (1928) 349-360. (with Whyburn, G. T.) 11. Continuous curves which are cyclioly connected. Bull. Acad. Polon. Sci..Math. (1928) 127-142. 12. Concerning subsets of a continuous curve, which can be connected through the complement of a continuous curve. Amer. J. Math. 50 (1928) 521-534. 186 CHAPTER 9

13. On continuous curves having certain properties. Proc. Nat. Acad. Sci. 15 (1929) 91-94. 14. On simple closed curves and open curves. ibid. 15 (1929) 94-96. 15. Conditions under which every arc of a continuous curve is a subset of a maximal arc of the curve. Math. Ann. 101 (1929) 194209. 16. Concerning the arc-curves and basic sets of a continuous curve (second paper). Trans. Amer. Math. Soc. 31 (1929) 595-612. 17. Continuous curves in which every arc may be extended. Bull. Amer. Math. Soc. 35 (1929) 850-858. 18. On continua which are disconnected by the omission of any point and some related problems. Monatsh. Math. Phys. 35 (1929) 135148. 19. Uber Verallgemeinerunger des Jordanschen Kontinuums. ibid. 36 (1929) 301-304. 20. Concerning continuous curves in metric space. Amer. J. Math. 51 (1929) 577-594. 21. Continuous curves homeomorphic with the boundary of a plane domain. Fund. Math. 14 (1929) 92-95. 22. On continuous images of a compact metric space. Fund. Math. 14 (1929) 334-338. 23. On the density of the cut points and end points of a continuum. Bull. Amer. Math. Soc. 36 (1930) 659-667. 24. A new proof of a theorem of Zarankiewicz. Fund. Math. 16 (1930) 134-135. 25. Some generalizations of the Scherrer Fixed-Point Theorem. ibid. 332-336. 26. On the regular points of a continuum. Trans. Amer. Math. Soc. 33 (1931) 252-262. 27. On avoidable points of continua with an application to end points. Math. Zeit. 34 (1931) 161-178. 28. A note on a property of continuous arcs. Proc. Camb. Phil. Soc. 27 (1931) 543-545. 29. On transformations having periodic properties. Fund. Math. 33 (1939) 95-103. 30. A note on the deﬁnition of arc-sets. Bull. Amer. Math. Soc. 35 (1940) 794-796. ACADEMIC DESCENDANTS AND PUBLICATIONS 187

31. A new proof of the cyclic connectivity theorem. Bull. Amer. Math. Soc. 48 (1942) 627-630.

Vance, Elbridge P. 03 • 1. Generalizations of nonalternating and non-separating transformations. Duke Math. J. 6 (1940) 66-79.

Ollman, L. F. 03 • 1. Solution of a problem of Ayres. Amer. J. Math. 64 (1942) 61-71.

Schnechenberger, Edith R. 03 • 1. On l-bounding monotonic transformations which are equivalent to homorphisms. Amer. J. Math. 63 (1941) 768-776.

Odle, John W. 03 • 1. Non-alternating and non-separating transformation modulo: A family of sets. Duke Math. J. 8 (1941) 256-268.

Miller, Don D. 03 • 1. Extension and reduction theorems for certain types of continuous transformations. Amer. J. Math. 64 (1942) 215-228. 2. Semigroups having zeroid elements. Amer. J. Math. 70 (1948) 117-125. 3. Regular D-classes in semigroups. Trans. Amer. Math. Soc. 82 (1956) 270-280. (with A. H. Cliﬀord)

Benton, Thomas C. 02 • 1. On continuous curves which are homogeneous except for a finite number of points. Fund. Math. 13 (1929) 151-177. 2. A definition of an unknotted simple closed curve. Bull. Amer. Math. Soc. 36 (1930) 406-408. 3. On continuous curves which are homogeneous except for a finite number of points (second part). Fund. Math. 15 (1930) 38-41. 4. Three-dimensional flows inside a cylinder. Quart. Appl. Math. 19 (1961-1962) 81-94. 188 CHAPTER 9

5. An example of the need for two stream functions in three dimensional ﬂows. Quart. Appl. Math. 21 (1963-1964) 235-237.

Zippin, Leo 02 • 1. On continuous curves and the Jordan curve theorem. Amer. J. Math. 52 (1930) 331-350. 2. A study of continuous curves and their relation to the JaniszewskiMul- likin theorem. Trans. Amer. Math. Soc. 31 (1929) 744-770. 3. On a problem of N. Aronsaza;n and an axiom of R. L. Moore. Bull. Amer. Math. Soc. 37 (1931) 276-280. 4. Generalization of a theorem due to C. M. Cleveland. Amer. J. Math. 54 (1932) 176-184. 5. The Moore-Kline problem. Trans. Amer. Math. Soc. 34 (1932) 705-721. 6. Independent arcs of a continuous curve. Ann. of Math. 34 (1933) 95-113. 7. A characterization of the closed 2-cell. Amer. J. Math. 55 (1933) 207-217. 8. On continuous curves irreducible about subsets. Fund. Math. 20 (1933) 197-205. 9. Semi-compact spaces. Amer. J. Math. 57 (1935) 327-341. 10. On a problem of Cech. Cas. Mat. a Fysiky 65 (1936) 49-52. 11. On monotonic, complete covering systems. Fund. Math. 27 (l936) 12. Transformation groups. Lectures in Topology (1941) 191-221. University of Michigan Press, Ann Arbor, ich. 13. Two-ended topological groups. Proc. Amer. Math. Soc. 1 (1950) 309-315 . 14. A theorem on the rotation of the two-sphere. Bull. Amer. Math. Soc. 46 (1940) 520-521. 15. Topological group foundations of rigid space geometry. Trans. Amer. Math. Soc. 48 (1940) 21-49. 16. A theorem on Lie groups. Bull. Amer. Math. Soc. 48 (1942) 448452 . 17. A class of transformation groups in En . Amer. J. Math. 65 (1943) 601-608. ACADEMIC DESCENDANTS AND PUBLICATIONS 189

18. Existence of subgroups isomorphic to the real numbers. Ann. of Math. (2) 53 (1951) 298-326. 19. Two-dimensional subgroups. Proc. Amer. Math. Soc. 2 (1951) 822838 . 20. Four-dimensional groups. Ann. of Math. (2) 55 (1952) 140-166. 21. Small subgroups of finite-dimensional groups. Ann. of Math. (2) 56 (1952) 213-241. 22. Small subgroups of finite-dimensional groups. Proc. Nat. Acad. Sci. U. S. A. 38 (1952) 440-442. 23. Examples of transformation groups. Proc. Amer. Math. Soc. 5 (1954) 460-465. 24. Topological transformation groups. Interscience Publishers, New York-London, 1955, xi+282 pp. 25. Uses of infinity. New Mathematical Library, 7. Random House, New York-Toronto, 1962, vii+151 pp. 26. Topological transformation groups. I. Ann. of Math. (2) 41 (1940) 778-791. (with Montgomery, Deane)

Rutt, N. E. 02 • 1. Concerning the cut points of a continuous curve when the arc curve, AB, contains exactly n independent arcs. Amer. J. Math. 51 (1929) 217-246. 2. On certain types of plane continua. Trans. Amer. Soc. 33 (1931) 806-810. 3. Concurrence and uncountability. Bull. Amer. Math. Soc. 39 (1939) 295-302. 4. Prime ends and order. Ann. of Math. 34 (1933) 416-440. 5. Some theorems on triodic continua. Amer. J. Math. 56 (1934) 122-132. 6. On derived sets. Nat. Math. Mag. 18 (1943) 53-63.

Claytol, W. Shieﬄin 02 • 1. Topological immersion of Peanian continua in a spherical surface. Ann. of Math. 35, No. 4 (1934) 809-835. 2. Peanian continua not imbeddable in a spherical surface. Ann. of Math. 38 (1937) 631-646. 190 CHAPTER 9

Milgram. Arthur N. 02 • 1. Partially ordered sets and topology. Proc. Nat. Acad. Sci. 26 (1940) 291-293. 2. Partially ordered sets and topology. Rep. Math. Colloquium (2) 2 (1940) 3-9. 3. On iterated mappings. Rep. Math. Colloquium (2) 2 (1940) 21-24. 4. On shortest paths through a set. Rep. Math. Colloquium (2) 2 (1940) 39-44. 5. A generalization of the Cauchy-Reiman equation. Rep. Math. Colloquium (2) 3 (1941) 28-30. 6. Some topologically invariant metric properties. Proc. Nat. Acad. Sci. 29 (1943) 193-195. 7. Extensions of coverings from subspaces to spaces. Rep. Math. Colloquium (2) 4 (1943) 16-21. 8. Some metric topological invariants. Rep. Math. Colloquium (2) 5-6 (1944) 25-35. 9. Bendpoints, geodesics, and free approximations of plane curves. Rep. Math. Colloquium (2) 7 (1946) 37-45. 10. Saturated polynomials. Rep. Math. Colloquium (2) 7 (1946) 65-67. 11. Multiplicative semigroups of continuous functions. Duke Math. J. 16 (1949) 377-383. 12. Decompositions and dimension of closed sets in R . Trans. Amer. Math. Soc. 43 (1938) 465-481. 13. A general existence theorem and some applications. Ann. of Math. 39 (1938) 804-810. 14. On shortest paths through a set.Reports of a Mathematical Col- loquium. 2 (1940) 39-44. 15. Parabolic equations. Partial Diﬀerential Equations. (Proc. Sum- mer Seminar, Boulder, Col., 1957) 327-339. Interscience, New York, 1964. Diﬀerential operators on Riemannian manifolds. Rend. Circ. Mat. Palermo (2) 2 (1953) 266-326 (1954). (with Aronszajn, Nathan) (with Lax, P. D.) ACADEMIC DESCENDANTS AND PUBLICATIONS 191

16. Parabolic equations. Contributions to the theory of partial differential equations.Ann. of Mathematics Studies, No. 33. Prince- ton Univ. Press, Princeton, N. J., 1954, pp. 167-190. (with Gallai, Tibor) 17. Verallgemeinerung eines graphentheoretischen Satzes von Redei. Acta Sci. Math. (Szeded) 21 (1960) 181-186. 18. On linear sets in metric spaces. Reports of a Mathematical Col- loquium 1 (1939) 16-17. (with Menger, K.) 19. Harmonic forms and heat conduction. I. Closed Riemannian manifolds. Proc. Nat. Acad. Sci. 37 (1951) 180-184. (with Rosenbloom, P. C.) 20. Heat conduction on Riemannian manifolds. H. Heat distribution on complexes and approximation theory. Proc. Nat. Acad. Sci. 37 (1951) 435-438.

Hemmingsen, Erik 02 • 1. On Weierstrass sums for integrals involving second derivatives. Rep. Math. Colloquium (2) 3 (1941) 31-33. 2. Some theorems in dimension theory for normal Hausdorﬀ spaces. Duke Math. J. 13 (1946) 495-504. 3. Plane continua admitting non-periodic autohomeomorphisms with equicontinuous interates. Math. Scand. 2 (1954) 119-141. 4. Open simplicial mappings of manifolds on manifolds. Duke Math. J. 32 (1965) 325-331. 5. Open simplicial mappings of spheres on spheres. Norske Vid. Selsk. Forh. (Trondheim) 40 (1967) 67-69. 6. Introduction to algebraic topology. Translated from the notes of Armin Thedy and Hel Braun by Erik Hemmingsen. Charles E. Merrill Publishing Co., Columbus, Ohio, 1969, ix+229 pp. 7. Light open maps on n-manifolds. Duke Math. J. 27 (1960) 527536. (with Church, Philip T.) 8. Light open maps on n-manifolds. II. Duke Math. J. 28 (1961) 607-623. 9. Light open maps on n-manifolds. III. Duke Math. J. 30 (1963) 379-389. 10. Lifting and projecting expansive homeomorphisms. Math. Sys- tems Theory 2 (1968) 7-15. (with Reddy, William L.) 192 CHAPTER 9

11. Expansive homeomorphisms on homogeneous spaces. Fund. Math. 64 (1969) 203-207.

Peterson, Bruce P. 03 • 1. Convexity of polyhedra. Illinois J. Math. 11 (1967) 330-335. 2. Relative convexity and total concavity. Illinois J. Math. 11 (1967) 616-627. 3. Formal contraction of the u-simplex. Canad. Math. Bull. 10 (1967) 659-664.

Reddy, William L. 03 • 1. The existence of expansive homeomorphisms on manifolds. Duke Math. J. 32 (1965) 627-632. 2. Lifting expansive homeomorphisms to symbolic ﬂows. Math. Systems Theory. 2 (1968) 91-92. 3. Almost periodic semigroups in transformation groups. Illinois J. Math. 12 (1968) 494-509. (with Coven, Ethan M.) 4. Limit set equivalences of replete semigroups. Proc. Amer. Math. Soc. 23 (1969) 625-630. 5. Lifting and projecting expansive homeomorphisms. Math. Sys- tems Theory 2 (1968) 7-15. (with Hemmingsen, E.) 6. Expansive homeomorphisms on homogeneous spaces. Fund. Math. 64 (1969) 203-207. (with Hemmingsen, E.)

Antonelli, Peter 03 • 1. Structure theory for Montgomery-Samelson fiberings between manifolds. I, II. Canadian J. Math. 21 (1969) 170-179; ibid. 21 (1969) 180-186. 2. On the stable diffeomorphism of homotopy spheres in the stable range, N¡2P. Bull. Amer. Math. Soc. 75 (1969) 343-346. 3. Montgomery-Samelson singular fiberings of spheres. Proc. Amer. Math. Soc. 22 (1969) 247-250.

Barrett, Lida K. 02 • 1. Regular curves and regular points of ﬁnite order. Duke Math. J. 22 (1955) 295-304. ACADEMIC DESCENDANTS AND PUBLICATIONS 193

2. On a question concerning partitioning raised by R. H. Bing. Proc. Amer. Math. Soc. 8 (1957) 602-603. 3. The structure of decomposable snakelike continua. Duke Math. J. 28 (1961) 515-522. 4. A note on regular curves. Proc. Amer. Math. Soc. 12 (1961) 603-606.

I Wiginton, C. Lamar 03 • 1. Factoring pointlike simplicial mappings. Trans. Amer. Math. Soc. 129 (1967) 344-359. 2. A characterization of the maximal subgroups of the semigroup of nxn complex matrices. Czech. Math. J. 18 (93) (1968) 675-677. (with Decell, Henry P. Jr.) 3. Spaces determined by a group of functions. Bull. Amer. Math. Soc. 74 (1968) 1110-1112. (with Shrader, Susan)

Miller, D. D. 03 • 1. and Clirford, A. H 2. Union and symmetry preserving indomorphisms of the semigroups of all binary relations on a set. Czech. Math. J. 20 (95) (1970) 303-314.

Reddy, W. L. 03 • 1. Montgomery Samelson coverings on spheres. Mich. Math. J. 17 (1970) 65-67. 2. Pointwise expansion homeomorphisms. J. London Math. Soc. (2) 2 (1970) 232-236. 3. Open simplicial maps of spheres on manifolds. Duke Math. J. 38 (1971) 137-145.

Blackwell, Paul 03 • 1. An alternative proof of a theorem of Erdos and Szekeres. Amer. Math. Monthly 78 (1971) 273.

Antonelli, Peter L. 03 • 1. Differentiable Montgomery-Samuelson fiberings with finite singular sets. Canadian J. Math. 21 (1969) 1489-1495. 194 CHAPTER 9

2. The non-ﬁnite type of some DiﬀoMn Bull. Amer. Math. Soc. 76 (1970) 1246-1250. (with Burghelea D; Kahn, P. J.)

Reddy, W. L. 03 • 1. Each compact orientable surface of positive genus admits an expansive homeomorphism. Paciﬁc J. Math. 35 (1970) 737-741. (with O’Brien, Thomas) ACADEMIC DESCENDANTS AND PUBLICATIONS 195 Publications of G. H. Hallett

Hallett, G. H. 01 • 1. Concerning the deﬁnition of a simple continuous arc. Bull. Amer. Math. Soc. 25 (1919) 325-326.

Publications of Anna M. Mullikan

Mullikin, Anna M. 01 • 1. Certain theorems relating to plane connected point sets. Trans. Amer. Math. Soc. 24 (1922) 144-162. 196 CHAPTER 9 R. L. Wilder and his Mathematical Descendants

01 Wilder, R. L. 02 Cohen , Leon W., University of Michigan , 1928 02 Swingle, Paul M., University of Michigan, 1929 02 Miller, Edwin W., University of Michigan, 1932? 02 Dancer, Wayne, University of Michigan, 1935 02 Vaughan , Herbert E ., University of Michigan , 1935 02 MacKay, Roy, University of Michigan, 1938 02 Kaplan, Samuel, University of Michigan, 1942 02 Butcher, Kay E., University of Michigan, 1946 (Mrs. George W. Whitehead) 02 Larguier , Everett H., S . J ., University of Michigan, 1947 02 Curtis , Morton L ., University of Michigan , 1950 03 Rice, Peter Milton, Florida State University, 1963 03 Chandler , Richard E., Florida State University , 1963 04Taylor , William Wallace , North Carolina State U., 1971 03 Greathouse, Charles A., Florida State University, 1963 03 Schaufele, Christopher B ., Florida State University , 1964 03 Morava, Jack, Florida State University, 1968 03 Morgan, John W., Florida State University, 1969 02 Dickinson, Mrs. Alice B., University of Michigan, 1952 02 Shoenﬁeld, Joseph R., University of Michigan, 1952 03 Anderson , David R., Duke University , 1962 03 Grilliot, Thomas J., Duke University, 1967 02 Roth, John P., University of Michigan, 1953 03 Merwin, R. E., University of Pennsylvania, 1965 02 Roberts , Robert A., University of Michigan , 1953 02 Brahana, Thomas R., University of Michigan, 1954 03 Patty, C. W., University of Georgia, 1960 03 Bell, Curtis P., University of Georgia, 1963 03 Rhee, Choon Jai, University of Georgia, 1965 03 McBay, Shirley M., University of Georgia, 1967 03 Reed, Bruce E., University of Georgia, 1968 02 Gary, John M., University of Michigan, 1956 02 Raymond, Frank A., University of Michigan, 1958 02 Kwun, Kyung Whan, Univeristy of Michigan, 1958 03 Boals, Alfred, Michigan State University, 1967 03 Tollefson, Jeﬀrey, Michigan State University, 1968 ACADEMIC DESCENDANTS AND PUBLICATIONS 197

03 Fremon, Richard, Michigan State University, 1969 03 Myung, Myung Mi, Michigan State University, 1970 03 Quinn, Joan, Michigan State University, 1970 03 Spence, Lawrence, Michigan State University, 1970 02 Kelley, John E., University of Michigan, 1960 02 Stoddard, James H., University of Michigan, 1961 02 Cross, Myrle V. Jr., University of Michigan, 1961 02 Lewis, Judith Ann, University of Michigan, 1966 198 CHAPTER 9 Publications of R. L. Wilder and his Mathematical Descendants

Wilder, R. L. 01 • 1. On the dispersion sets of connected point sets. Fund. Math. 6 (1924) 214-228. 2. A theorem on continua. ibid. 7 (1925) 311-313. 3. Concerning continuous curves. ibid. 340-377. 4. A property which characterizes continuous curves. Proc. Na.. Acad. Sci. 11 (1925) 725-728. 5. A theorem on connected point sets which are connected im leinen. Bull. Amer. Math. Soc. 32 (1926) 338-340. 6. A connected and regular point set which has no sub-continuum. Trans. Amer. Math. Soc. 29 (1927) 332-340. 7. A point set which has no true quasi-components, and which becomes connected upon the addition of a single point. Bull. Amer. Math. Soc. 33 (1927) 432-427. 8. The non-existence of a certain type of regular point set. Bull. Amer. Math. Soc. 33 (1927) 439-446. 9. On connected and reglzr point sets. ibid. 34 (lg28) 649-655. 10. Concerning R. L. Moore’s axioms for plane analysis situs. ibid. 752-760. 11. A characterization of continuous curves by a property of their open subsets. Fund. Math. 11 (1928) 127-131. 12. On a certain type of connected set which cuts the plane. Proc. Internat. Math. Congress in Toronto 1 (1928) 423-437. 13. Concerning zero-dimensional sets in Euclidean space. Trans. Amer. Math. Soc. 31 (1929) 354-359. 14. Characterizations of continuous curves that are perfectly continuous. Proc. Nat. Acad. Sci. 15 (1929) 614-621. 15. Concerning perfect continuous curves. ibid. 16 (1930) 223-240. 16. A converse of the Jordan-Brower separation theorem in three dimensions. Trans. Amer. Math. Soc. 32 (1930) 632-657. 17. Concerning simple continuous curves and related point sets. Amer. J. Math. 53 (1931) 38-55. ACADEMIC DESCENDANTS AND PUBLICATIONS 199

18. Extension of a theorem of Mazurkiewicz. Bull. Amer. Math. Soc. 37 (1931) 287-293. 19. A plane arcwise connected im kleinen point set which is not strongly connected im kleinen. ibid. (1932) 531-532. 20. Point sets in three and higher dimensions and their investigation by means of a uniﬁed analysis situs. ibid. 649-692. 21. On the imbedding of subsets of a metric space in Jordan continua. Fund. Math. l9 (1932) 45-64. 22. On the linking of Jordan continua in En by (n - 2) cycles. Ann. of Math. 34 (1933) 441-449. 23. Concerning a problem of K. Borsuk. Fund. Math. 21 (1933) 156-167. 24. On the properties of domains and their boundaries in En. Math. Ann. 109 (1933) 273-306. 25. Concerning irreducible connected sets and irreducible regular connexes. Amer. J. Math. 56 (1934) 547-557. 26. Generaiized closed manifolds in n-space. Ann. of Math. 35 (1934) 876-903. 27. On free subsets of En. Fund. Math. 25 (1935) 200-208. 28. On locally connected spaces. Duke Math. J. I (1935) 543-555. 29. A characterization of manifold boundaries in En dependent only on lower dimensional connectivities of the complement. Bull. Amer. Math. Soc. 42 (1936) 436-441. 30. The strong symmetrical cut-sets of closed Euclidean n-space. Fund. Math. 27 (1936) 136-139. 31. The sphere in topology. American Mathematical Society Semi- centennial Publications. V. 2 (1968) 136-184. 32. Sets which satisfy certain avoidability conditions. Cas. Pest. Mat. a Fysiky 67 (1937-38) 185-198.

33. Property Sn. Amer. J. Math. 61 (1939) 823-832. 34. Decompositions of compact metric spaces. Amer. J. Math. 63 (1941) 691-697. 35. Uniform local connectedness. Lectures in Topology. pp. 29-41. University of Michigan Press, Ann Arbor, Michigan, 1941. 36. Topology of manifolds. Amer. Math. Soc. Colloquium Publica- tions, V. 32 (1949), 5+402 pp. 200 CHAPTER 9

37. Introduction to the foundations of mathematics. John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1952 xiv+305 pp. 38. The origin and growth of mathematical concepts. Bull. Amer. Math. Soc. 59 (1953) 423-448. 39. A type of connectivity. Proc. Inter. Cong. Mathematicians. Amsterdam 3 (1954) 264-64. (Abstract) 40. Concerning a problem of Alexandroff. Mich. Math. J. 3 (1955) 181-185. 41. Some consequences of a method of proof of J. H. C. Whitehead. Mich. Math. J. 4 (1957) 27-31. 42. Some mapping theorems, with applications to non-locally connected spaces. Algebraic geometry and topology. A symposium in honor of S. Lefschetz. pp. 378-388. Princeton University Press, Princeton, New Jersey, 1957. 43. Monotone mappings of manifolds. Pacific J. Math. 7 (1957) 1519-1528. 44. Monotone mappings of manifolds. II. Mich. Math. J. 5 (1958) 19-23. 45. Local orientability. Colloq. Math. 6 (1958) 79-95. 46. Axiomatics and the development of creative talent. The Ax- iomatic Method. Ed. by L. Henkin, P. Suppes, and A. Taraki. Amsterdam (1959) 474-488. 47. The nature of modern mathematics (Russel Lecture for 1959). Mich. Alumnus Quarterly Review. 65 (1959) 302-312. 48. Mathematics: A cultural phenomenon. In Essays in the Science of Culture. Ed. by G. E. Dole and R. L. Carneiro, New York (1960) 471-485. 49. A certain class of topological properties. Bull. Amer. Math. Soc. 66 (1960) 205-239. 50. Extension of local and medial properties to compactifications with an application to Cech manifolds. Czech. Math. J. 11 (1961) 306-310. 51. A converse of a theorem of R. H. Bing and its generalizations. Fund. Math. 50 (1961) 119-122. 52. Material and method, in Undergraduate Research in Mathemat- ics. Ed. v K. O. May and S. Schuster, Northfield, Minn. (1961) 9-27. ACADEMIC DESCENDANTS AND PUBLICATIONS 201

53. Freeneso in n-space, in Topology of 3-manifolds and Related Top- ics, M. K. Fort, ed., New York, Prentice-Hall (1962) 106-109. 54. Partially free subsets of Euclidean n-space. Mich. Math. J. 9 (1962) 97-107. 55. Topology: Its nature and significance. The Math. Teacher 55 (1962) 462-475. 56. A problem of Bing. Proc. Nat. Acad. Sci. 54 (1965) 683-687. 57. An elementary property of closed coverings of manifolds. Mich. Math. J., 13 (1966) 49-55. 58. role of the axiomatic method. Amer. Math. Monthly 74 (1967) 115-127. 59. The nature and role of research in mathematics, in Research: Definitions and Reflections (Essays on the occasion of the Uni versity of Michigan’s Sesquicentennial), Univ. of Michigan, Ann Arbor, Michigan (1967) 96-109. 60. The role of intuition. Science. 156 (1967) 605-610. 61. Addition and reduction theorems for medial properties. Trans. Amer. Math. Soc. 130 (1968) 131-140. 62. Mathematics’ Biotic Origins. Medical Opinion and Review. 5 (1969) 124-135. 63. nds and social implications of research. Bull. Amer. Math. Soc. 75 (1969) 891-906. 64. Development of modern mathematics, in Historical Topics for the Mathematics Classroom, 31st Yearbook, Nat’l Council of Teachers of Math., Washington, D. C. (1969) 460-476. 65. The nature of research in mathematics, in The Spirit and Uses of the Mathematical Sciences, McGraw-Hill, New York (1969) 31-47. 66. The beginning teacher of college mathematics, in Effective Col- lege Teaching, Amer. Council on Education, Washington, D. C. (1970) 94-103. 67. Historical background of innovations in mathematics curricula in Mathematics Education , 69th Yearbook of Nat’l. Soc. for the Study of Educ., Part I, Univ. of Chicago Press (1970) 7-22. 68. The beginning teacher of college mathematics. CUPM Newslet- ter. December, No. 6 (1970). 202 CHAPTER 9

69. The existence of certain types of manifolds. Trans. Amer. Math. Soc.. 91 (1959) 162-169. (with Curtis, M. L.) 70. On certain inequalities relating the Betti numbers of a manifold and its subsets. Proc. Nat. Acad. Sci. 40 (1954) 207-209. (with Roth, J. P.)

Cohen, L. W. 02 • 1. On the mean ergodic theorem. Ann. of Math. (2) 41, 505-509 (1940). 2. On topological completeness. Bull. Amer. Math. Soc. 46, 706-610 (1940) 3. Uniformity in topological space. Lectures in Topology, pp. 255- 265. University of Mich. Press, Ann Arbor, Mich., 1941. 4. On linear equations in Hilbert space. Bull. Amer. Math. Soc. 50, 729-733 (1944). 5. A non-Archimedian measure in the space of real sequences. Pa- cific J. Math. 6 (1956) 9-24. 6. A theory of transfinite convergence. Trans. Amer. Math. Soc. 66 (1949) 65-74. (with Goffman, Casper.) 7. The topology of ordered Abelian groups. Trans. Amer. Math. Soc. 67 (1949) 310-319. 8. On completeness in the sense of Archimedes. Amer. J. Math. 72 (1950) 747-751. 9. On completeness and category in uniform space. Amer. J. Math. 72 (1950) 752-756. 10. On the metrization of uniform space. Proc. Amer. Math. Soc. (1950) 750-753

Single, P. M. 02 • 1. An unnecessary condition in two theorems of analysis situs. Bull. Amer. Math. Soc. 34 (1928) 607-618. 2. Certain type of continuous curve and related point sets. Trans. Amer. Math. Soc. 33 (1929) 544-556. 3. Generalized indecomposable continua. Amer. J. Math. 52 (1930) 647-658. 4. Generalizations of bioconnected sets. ibid. 53 (1931) 385-400. ACADEMIC DESCENDANTS AND PUBLICATIONS 203

5. Two types of connected sets. Bull. Amer. Math. Soc. 37 (1931) 254-258. 6. End-sets of continua irreducible between two points. Fund. Math. 17 (1931) 40-76. 7. Bioconnected and related sets. Amer. J. Math. 54 (1932) 525- 535. 8. A ﬁnitely-containing connected set. Bull. Amer. Math. Soc. 46 (1940) 178-181. 9. Indecomposable connexes. ibid. 47 (1941) 796-803. 10. The closure of types of connected sets. Proc. Amer. Math. Soc. 2 (1951) 178-185. 11. Local properties and sums of trajectories. Portugal. Math. 15 (1956) 89-103. 12. Sums of sets with indecomposable properties. ibid. 16 (1957) 129-144. 13. Higher dimensional indecomposable connected sets. Proc. Amer. Math. Soc. 8 (1957) 816-819. 14. Connected sets of Van Vleck. ibid. 9 (1958) 477-482. 15. Algebras and connected sets of Vitali. Portugal. Math. 18 (1959) 69-85. 16. The existence of widely connected and biconnected semigroups. Proc. Amer. Math. Soc. 11 (1960) 243-248. 17. Connected sets of Wada. Mich. Math. J. 8 (1961) 77-95. 18. Extended topologies and iteration and recursion of set-functions. Portugal. Math. 23 (1964) 103-129. (with Davis, Harvey S.) 19. Semigroups, continua and the set functions Tn. Duke Math. J. 29 (1962) 265-280. (with Davis, H. S.; Stadtlander, D. P.) 20. Properties of the set functions T . Portugal. Math. 21 (1962) 113-133. 21. Characterization of n-spheres by an excluded middle membrane principle. Mich. Math. J. 11 (1964) 53-59. (with Dickman, R. F.; Rubin, L. R.) 22. Another characterization of the n-sphere and related results. Pa- ciﬁc J. Math. 14 (1964) 871-878. 23. Irreducible continua and generalization of hereditarily unicoherent continua by means of membranes. J. Austral. Math. Soc. 5 (1965) 416-426. Quoting from the paper in Mich. Math. J. 11 (1964) 53-59 (MR28 #4523). 204 CHAPTER 9

24. Semigroups and clusters of indecomposability. Fund. Math. 56 (1964) 21-23. (with Dickman, R. F.; Kelley, l. L. and Rubin, L. R.) 25. Core decompositions of continua. Fund. Math. 61 (1967) 33-50. (with Fitzgerald, R. W.) 26. Indecomosable trajectories. Tohoku Math. J. (2) 10 (1958) 3-10. (with Hunter, Robert P.) 27. Concerning minimal generating subsets of semigroups. Portugal. Math. 20 (1961) 231-250. (with Kelley, J. L.)

Miller, E. W. 02 • 1. On subsets of a continuous curve which lie on an arc of the continuous curve. Amer. J. Math. 54 (1932) 397-416. 2. On certain properties of Frechet L-spaces. Fund. Math. 26 (1936) 116-119. 3. On a property of families of sets. Comptes Rendus des Seances des Sciences et des Lettres de Varsovie, Classe II, 30 (1937) 31- 38. 4. Some theorems on continua. Bull. Amer. Math. Soc. 46 (1940) 150157. 5. A note on Souslin’s problem. Amer. J. Math. 65 (1943) 673-678. 6. Concerning similarity transformations of linearly ordered sets. Bull. Amer. Math. Soc. 46 (1940) 322-326. (with Dushnik, Ben) 7. Partially ordered sets. Amer. J. Math. 63 (1941) 600-610. 8. Zero-dimensional families of sets. Bull. Amer. Math. Soc. 47 (1941) 921-923. (with Eilenberg, Samuel)

Vaughan, H. E. 02 • 1. On a class of metrics deﬁning a metrizable space. Bull. Amer. Math. Soc. 44 (1938) 557-561. 2. On locally bicompact spaces. Fund. Math. 31 (1938) 15-21. 3. Well-ordered subsets and maximal members of ordered sets. Pa- ciﬁc J. Math. 2 (1952) 407-412. 4. On two theorems of plane topology. Amer. Math. Monthly 60 (1953) 462-468. ACADEMIC DESCENDANTS AND PUBLICATIONS 205

5. Characterization of the sine and cosine. ibid. 62 (1955) 707713. 6. Cliques and groups. Math. Gaz. 52 (1968) 347-350. 7. Hyperspheres associated with an n-simplex. Amer. Math. Monthly 74 (1967) 384-392. (with Gabai, Hyman) 8. The marriage problem. Amer. J. Math. 72 (1950) 214-215. (with Halmos, Paul R.)

Kaplan, Samuel 02 • 1. Homology properties of arbitrary subsets of Euclidean spaces. Trans. Amer. Math. Soc. 62 (1947) 248-271. 2. A zero-dimensional topological group with a one-dimensional factor group. Bull. Amer. Math. Soc. 54 (1948) 964-968. 3. Extensions of the Pontrjagin duality. I. Inﬁnite products. Duke Math. J. 15 (1948) 649-658. 4. Extensions of Pontrjagin duality. II. Direct and inverse sequence Duke Math. J. 17 (1950) 419-435. 5. Cartesian products of reals. Amer. J. Math. 74 (1952) 936-954. 6. Biorthogonality and integration. Proc. Amer. Math. Soc. 7 (1956) 109-114. 7. On the second dual of the space of continuous functions. Trans. Amer. Math. Soc. 86 (1957) 70-90. 8. The second dual of the space of continuous functions. II. Trans. Amer. Math. Soc. 93 (1959) 329-350. 9. The second dual of the space of continuous functions. III. Trans. Amer. Math. Soc. 101 (1961) 34-51. 10. The second dual of the space of continuous functions. Trans. Amer. Math. Soc. 113 (1964) 512-546. 11. Closure properties of C(X) in its second dual. Proc. Amer. Math. Soc. 17 (1966) 401-406. 12. The second dual of the space of continuous functions and the Riemann integer. Illinois J. Math. 12 (1968) 283-302.

Larguier, Everett H. 02 • 1. A matrix theory of n-dimensional measurement. Duke Math. J. 5 (1939) 729-739. 2. Postulational methods. Scripta Math. 8 (1941) 99-109. 206 CHAPTER 9

3. Homology bases with applications to local connectedness. Paciﬁc J. Math. 2 (1952) 191-208.

Curtis, Morton L. 02 • 1. Deformation-free continua. Ann. of Math. (2) 57 (1953) 231-247 2. Classification spaces for a class of fiber spaces. Ann. of Math. Vol. 60 (1954) 304-316. 3. A note on monotone deformation-free mappings. Proc. Amer. Math. Soc. 5 (1954) 437-438. 4. The covering theorem. Proc. Amer. Math. Soc. 7 (1956) 682- 684 5. An imbedding theorem. Duke Math. J. 24 (1957) 349-351. 6. A note on Kosinski’s r-spaces. Fund. Math. 46 (1958) 25-27. 7. elf-linked subgroups of semigroups. Amer. J. Math. 81 (1959) 889-892. 8. Homotopically homogeneous polyhedra. Mich. Math. J. 8 (1961) 55-60. 9. Cartesian products with intervals. Proc. Amer. Math. Soc. 12 (1961) 819-82. 10. Sllrinking continua in 3-space. Proc. Cambridge Philos. Soc. 57 (1961) 432-433. 11. Homotopy manifolds. Topology of 3-manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) 102-104. Pren- ticeHall, Englewood Cliffs, N. J., 1962. 12. On 2-complexes in 4-space topology of 3-manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) 204-207. PrenticeHall, Englewood Cliffs, N. J., 1962. 13. Smoothness conditions on continua in Euclidean space. Mich. Math. J. 14 (1967) 277-281. 14. Knotted 2-spheres in the 4-sphere. Ann. of Math. (2) 70 (1959) 565-571. (with Andrews, J. J.) 15. n-space modulo an arc. Ann. of Math. (2) 75 (1962) 1-7. 16. Free groups and handle-bodies. Proc. Amer. Math. Soc. 16 (1965) 192-195. (with Andrews, J. J.) 17. Etended Nielsen operations in free groups. Amer. Math. Monthly 73 (1966) 21-28. ACADEMIC DESCENDANTS AND PUBLICATIONS 207

18. Imbedding decompositions of E3 in E4. Proc. Amer. Math. Soc. 11 (1960) 149-155. (with Bing, R. H.) 19. Groups which are cogroups. Proc. Amer. Math. Soc. 22 (1969) 235-237. (with Dugundji, James) 20. Certain subgroups of the homotopy groups. Mich. Math. J. 4 (1957) 167-172. (with Fort, M. K., Jr.) 21. Homotopy groups of one-dimensional spaces. Proc. Amer. Math. Soc., 8 (1957) 577-579. 22. Singular homology of one-dimensional spaces. Ann. of Math. (2) 69 (1959) 309-313. 23. The fundamental group of one-dimensional spaces. Proc. Amer. Math. Soc. 10 (l959) 140-148. 24. lnfinite sums of manifolds. Topology 3 (1965) 31-42. (with XXXliln, W.) 25. Homotopy equivalence of fiber bundles. Proc. Amer. Math. Soc. 9 (1958) 178-182. (with R. Lashof.) 26. On product and bundle neighborhoods. Proc. Amer. Math. Soc. I3 (1962) 934-937. 27. Celluarity of sets in products. Mich.. Math. J. 9 (196) 299-30. (with McMillan, D. R.) 28. The existence of certain types of manifolds. Trans. Amer. Math. Soc. 91 (1959) 152-160. (with Wilder, R. L.) 29. A theorem on dimension. Proc. Amer. Math. Soc. 3 (1952) 159-161. (with Young, Gail S.) 30. On the polyhedral Schoenflies theorem. Proc. Amer. Math. Soc. 11 (196) 888-889. (with Zeeman, E. C.) 31. Mislin, Guido 32. Two new H-spaces. Bull. Amer. Math. Soc. 76 (1970) 851-852. 33. A proof of deRham’s theorem. Fund. Math. 68 (1970) 265-268. (with Dugundji, J.) 34. Finite dimensional H-spaces. Bull. Amer. Math. Soc. 77 (1971) 1-12

Rice Peter Milton 03 • 1. The Hauptvermutung and the polyhedral Schoenﬂies theorem. Bull. Amer. Math. Soc. 71 (1965) 521-522. 208 CHAPTER 9

2. Homotopically homogeneous spaces and manifolds. Trans. Amer. Math. Soc. 120 (1965) 247-254. 3. On manifold-like polyhedra. Mich. Math. J. 13 (1966) 375-376. 4. Taming knots. Proc. Amer. Math. Soc. 19 (1968) 254. 5. r ﬂows on R3. Math. Scand. 21 (1967) 128-135 (1968). 6. Actions on Z4 in S3. Duke Math. J. 36 (1969) 749-751.

Greathouse, Charles A. 03 • 1. Flat, locally tame, and tame embeddings. Bull. Amer. Math. Soc. 69 (1963) 820-823. 2. Locally ﬂat strings. Bull. Amer. Math. Soc. 70 (1964) 415-418. 3. The equivalence of the annulus conjecture and the slab conjecture. Bull. Amer. Math. Soc. 70 (1964) 716-717. 4. llullrity at the boundary of a manifold. Proc. Amer. Math. Soc. ii (1965) 1334-1341.

Schaufele, C. B. 03 • 1. on Link groups. Bull. Amer. Math. Soc. 72 (196) 107-110. 2. Kernals of free abelian representations of a link group. Proc. Amer. Math. Soc. 18 (1967) 535-539. 3. The commutator group of a doubled knot. Duke Math. J. 34 (1967) 677-681.

Dickinson, Alice 02 • 1. ”S’7 condition, and uniform structures. Amer. .J. Math. 75 !1XXX ’4-228.

Shoenﬁeld, Joseph R. 02 • 1. Relative consistency proof. J. Symbolic Logic 19 (1954) 21-28. 2. The structure of locally compact groups. Mathematics Depart- ment, Duke University, Durham, North Carolina, 1956. 66 pp. 3. i-creative sets. Proc. Amer. Math. Soc. 8 (1957) 964-967 4. Functionelles recursivement deﬁnissables et functionelles slves.C. R. Acad. Sci. Paris 245 (1957) 399-402. (with Kreisel, Georg; Lacombe, Daniel) ACADEMIC DESCENDANTS AND PUBLICATIONS 209

5. On theoretic concepts and recursive well-orderings. Arch. de Logi. Grundlagenforsch. 5 (1960) 42-64. (with Kreisel Georg; Wang Hao) 6. A hierarchy baset on a type two object. Trans. Amer. Math. Soc. 134 (1968) 103-108. 7. Unramiﬁet forcing. Axiomatic Set Theory (Proc. Sympos. Pure Math. Vol. XIII part 1, Los Angeles, Calif., 1967) 357-381. American Mathematical Society, Providence, Rhode Island, 1971.

Anderson, David R. 03 • 1. st2nee theorem for a solution of ﬂ()=F(x,f(g(x)). SIAM Rev. 3 (1966) 359-362.

Elliot Thomas J. 03 • 1. Extensions of algebra homomorphisms. Mich. Math. J. 14 (1967)

Ji, John T. 02 • 1. Applications of algebraic topology to numerical analysis: on the siutin t t trL prLlLm. L Nat. Acad. Sci. (1955) 518-5XX1. 2. ’l JaLiditJ of Kron’s method of tearing. ibid. 41 (1955) 599-600. 3. Algebraic topological methods for the synthesis of switching systems. Trans. Amer. Math. Soc. 88 (1958) 301-326. 4. -brl, topological methods in synthesis. Proc. Internat. Sympos. XXXXXXX Harvard Univ. Press. 5. Icati n algbraic topology: Kron’s method of tearing. Quart. Appl. Math. 17 (1959) 1-24. 6. Minimization over Boolean trees. IBM J. Res. Develop. 4 (1960) 543-558. 7. Diagnosis of automata failures: A calculus and a method. IBM J. Res. Develop. 10 (1966) 278-291. 8. Programmed algorithms to compute tests to detect and distin- guish between failures in logic circuits. IEEE Trans. Electronic Computers EC-16 (1967) 567-580. (with Bouricius, Willard G., Schneider, Peter R.) 9. Minimization over Boolean graphs. IBM J. Res. Develop. 6 (1962) 227-238.(with Karp, R. M.) 210 CHAPTER 9

10. A rector method for solving linear equations and inverting matrices. J. Math. Phys. 35 (1956) 312-317. (with Scott, D. S.) 11. Algebraic topological methods for the synthesis of switching systems. III. Minimization of nonsingular Boolean trees. IBM J. Res. Develop. 3 (1959) 326-344. (with Wagner, E. C.) 12. On certain inequalities relating the Betti numbers of a manifold and it subsets. Pro. Nat. Acad. Sci. 40 (1954) 207-20. (with Wilder, R. L.)

Brahana Thomas R. 02 • 1. Products of quasi-complexes. Proc. Amer. Math. Soc. 7 (1956) 954-95. 2. A theorem about Betti groups. Mich. Math. J. 4 (1957) 33-37. 3. Axioms for local homology theory. Duke Math. J. 25 (1958) 381-399 4. Products of generalized manifolds. Illinois J. Math. 2 (1958) 76-80. 5. On a class of isotopy invariants. Topology of 3-manifolds and related topics (Proc. University of Georgia Institute, 1961) 235- 237. Prentice-Hall, Englewood Cliﬀs, N. J., 1962. 6. mXXXXldi regular neighborhod of the singular locus of a 2 dimensional polyhedroll in 1.3. Duke Math. J. 30 (196) 215-220. 7. Homotopy for cellular set-valued functions. Proc. Amer. Math. Soc. 16 (1965) 455-459. (with Fort M. K., Tlorstman, Walt C.)

Patty, C. Wayne 03 • 1. Homotopy groups of certain deleted product .spaces. Proc. Amer. Math. Soc. 12 (1961) 369-373. 2. The homology of deleted products of trees. Duke Math. J. 29 (1962) 413-428. 3. The fundamental group of certain deleted product spaces. Trans. Amer. Math. Soc. 105 (1962) 314-321. 4. A note on the homology of deleted product spaces. Proc. Amer. Math. Soc. 14 (1963) 800. 5. Isotopy invariants of trees. Duke Math. J. 31 (1964) 183-197. 6. Isotopy classes of imbeddings. Trans. Amer. Math. Soc. 128 (1967) 232-247. ACADEMIC DESCENDANTS AND PUBLICATIONS 211

7. Homology of deleted products of contractible 2-dimensional polyhedra. I. Canad. J. Math. 20 (1968) 416-44,1. 8. Homology o deleted products of contractible 2 -dimensional polyhedra. II. Canadian J. Math. 20 (1968) 842-854. 9. Polyhedra whose deleted products have tht llomotopy type o the n-spheres. Duke Math. J. 36 (1969) 233-236. 10. Deleted products with homotopy types of spheres. Trans. Amer. Math. Soc. 147 (1970) 223-240. 11. The comparison of topologies. Duke Math. J. 36 (1969) 325-331. (with Fletcher, Peter; Hoyle, Hughes B., III.) 12. Homology of teletet protucts of one-dimensional spaces. Trans. Amer. Math. Soc. 151 (1970) 499-510.(with Copeland, Arthur H., Jr.) 13. Homology of deleted products of contractible 2-dimensional poly- hetral. III Canadian J. Math. 22 (1971) 1217-1223.

Rhee, Choon Jai 03 • 1. Homotopy groups for cellular set-valued functions. Proc. Amer. Math. Soc. l9 (1968) 874-876. 2. A note on the Riemann mapping theorem. Kyungpook Math. J. 8 (1969) 47-48. 3. A short note on the Riemann mapping theorem. Kyungpook Math. J. 8 (198) 49-50. 4. Cohomotopy groups of closed bounded subsets in Banach spaces. Bull. Soc. Roy. Sci. Liege 37 (1968) 381-386. (with Pak, J.)

Reed, Bruce E. 03 • 1. Representations of solvable Lie algebras. Mich. Math. J. 16 (1969) 27-233.

Gary, John M. 02 • 1. Higher dimensional cyclic elements. Pacific J. Math. 9 (1959) 1061-1070. 2. The topological structure of trajectories. Mich. Math. J. (1960) xxxx. 3. On certain finite difference schemes for hyperbolic systems. Math. Comp. 18 (1964) 1-18. 212 CHAPTER 9

4. Hyman’s method applied to the general eigenvalue problem. Math. Comp. 19 (1965) 314-316. 5. Computing eignevalues of ordinary differential equations by finite differences. Math. Comp. 19 (1965) 365-379. 6. A generalization of the Lax-Richtmyer theorem on finite difference schemes. SIAM J. Numer. Anal. 3 (1966) 467-473. 7. On convergence rates for line overrelaxation. Math. Comp. 21 (1967) 20-23. 8. A matrix method for ordinary differential eigenvalue problems. J. Computational Phys. 5 (1970) 169-187.(with Helgason, Richard)

Raymond, Frank A. 02 • 1. A note on the local “C” groups of Griffiths. Mich. Math. J. 7 (1960) 1-5. 2. Separation and union theorems for generalized manifolds with boundary. Mich. Math. J. 7 (1960) 7-21. 3. The end point compactification of manifolds. Pacific J. Math. 10 (1960) 947-963. 4. Local cohomology groups with closed supports. Math Zeit. 76 (1961) 31-41. 5. The orbit spaces of totally disconnected groups of tranformations on manifods. Proc. Amer. Math. Soc. 12 (1961) l-7. 6. Local triviality for Hurewicz fiberings of manifolds. Topology 3 (1965) 43-57. 7. Some remarks on the co-efficients xxxxx in the theory of lonoloy manifolds. Pacific J. Math. 15 xxxxxxxx. 8. Two problems in the theory of generalized manifolds. Mich. Math. J. 14 (1967) 353-356. 9. Classification of the actions of the circLe on 3-manifold.s. Trans. Amer. Math. Soc. 131 (l968) 61 -78. 10. Exotic PL actions which are topologically linear. Proc . Conf . on Transformation Groups (New Orleans, La., l967) pp 339-340 Springer, New York, 1968. 11. p-adic groups of transformations. Trans. Amer. Math. Soc. 99 (1968) 488-498. (with C. E. Bredon; Williams , R. F.) 12. Generalized cells in generalized manifolcis. Proc. Amer. Math. Soc. 11 (1960) 135-139. (with Kwun, K. W.) ACADEMIC DESCENDANTS AND PUBLICATIONS 213

13. Factors of cubes. Amer. J. Math. 84 (1960) 433-440. 14. Mapping cylinder neighborhoods. Mich. Math. J. 10 (1963) 353-357 . 15. Manifolds which are joins. Trans. Amer. Math. Soc. 111 (1964) 108-120 . 16. Almost acyclic maps of manifolds. Amer. J. Math. 86 (1964) 638-650 . 17. Spherical manifolds. Duke Math. J. 34 (1967) 397-401. (with and Kwun, K. W.) 18. Czech extensions of contraviant functors. Trans. Amer. Math. Soc. 133 (1968) 415-434. (with Lee, C. N.) 19. Examples of p-adic transformation groups. Ann. of Math. (2) 78 (1963) 92- 106 . 20. Cohomological and dimension theoretical properties of orbit spaces of p-adic actions. Proc. Conf. on Transformation Groups (New Orleans, Louisiana, 1967) 354-365. Springer, New York, 1968. 21. Actions of S0(2) on 3-manifolts. ibid. 297-318. (with Orlik, Peter 22. Actions of compact Lie groups on aspherical manifolts. ibit. 227-264. (with Conner, P. E. 23. Inective operations of the toral groups. Topology 10 (1971) 283- 296. 24. On 3-manifolts with local S0(2) action. Quart. J. Math. Oxford Ser. (2) 20 (1969) 143-160. (with Orlik, Peter) 25. Actions of the torus on 4-manifolds. I. Trans. Amer. Math. Soc. 152 (1970) 531-559.

Kwun, Kyung Whan 02 • 1. A generalized manifold. Mich. Math. J. 6 (1959) 299-302. 2. A fundamental theorem on decompositions of th( plere into points and tame arcs. Proc. Amer. Math. Soc. 12 (1961) 47-50. 3. A characterization of the n-sphere. Trans. Amer. Math. Soc. 101 (1961) 377-383. 4. Factors of N-space. Mich. Math. J. 9 (1962) 207-211 5. Upper semicontinuous decompositions of the n-sphere. Proc. Amer. Math. Soc. 13 (1962) 284-290. 214 CHAPTER 9

6. On 3-manifolds that are not simply connected. Proc. Amer. Math. Soc. 13 (1962) 291-292. 7. An involution of the n-cell. Duke Math. J. 30 (1963) 443-446. 8. Product of Euclidean spaces with an arc. Annals. of Math.. xxxx9 (1964) 14-1xxx. 9. Uniqueness of the open cone neighborhood. Proc. Amer. Math. Soc. 15 (1964) 476-479. 10. Factors of cubes. Amer. J. Math. 84 (1962) 433-440. 11. Mapping cylinder neighborhoods. Mich. Math. J. 10 (1963) 353-357. 12. Manifolds which are joins. Trans. Amer. Math. Soc. 11 (1964) 108-120. 13. Examples of generalized manifold approaches to topological manifolds. Mich. Math. J. 14 (1967) 225-229. 14. Open manifolds with monotone union property. Proc. Amer. Math. Soc. 17 (1966) 1091-1093. 15. Compactiﬁcations of n-space by an arc. Proc. Amer. Math. Soc. 19 (1968) 1133-1137. 16. Piecewise linear involutions of S1 S2. Mich. Math. J. 16 (1969) 93-96. × 17. Involutions of the n-cell. Proc. Conf. on Tranformation Groups (New Orleans, La., 1967) pp. 343-344 Springer, New York, 1968. 18. Nonexistence of orientation reversing involutions on some manifolds. Proc. Amer. Math. Soc. 23 (1969) 725-726. 19. 3-Manifolds which double-cover themselves. Amer. J. Math. 91 (1969) 441-452. 20. Inﬁnite sums of manifolds. Topology 3 (1965) 31-42. (with Cur- tis, M. L.) 21. Shrinking a manifold in a manifold. Proc. Nat. Acad. Sci. (1966) 259-261. (with Hocking, J. G.) 22. eneralized cells in generalized manifolds. Proc. Amer. Math. Soc. ll (1960) 135-139. (with Raymond, Frank) 23. Almost acylic maps of manifolds. Amer. J. Math. 86 (1964) 650. 24. Spherical manifolds. Duke Math. J. 34 (1967) 397-401. ACADEMIC DESCENDANTS AND PUBLICATIONS 215

25. Product and sum theorems for Whitehead torsion. Ann. of Math. (2) 82 (1965) 183-190. (with Szczarba, R. H.) 26. scarcity of orientation-reversing PL involutions of lens spaces. Mich. Math. J. 17 (1970) 355-358.

Tollefson, Jeﬀrey 03 • 1. xxxmanfolds that cover themselves. Mich. Math. J. 16 (1969) 103-109. 2. Free involutions on non-prime 3-manifolds. Osaka J. Math. 7 (1970) 161-164. 3. A characterization of 3-manifolds that are protucts Amer. J. Math. 92 (1970) 604-611. 4. Imbedding free cyclic group actions in circle group actions. Proc. Amer. Math. Soc. 26 (1970) 671-673. (with Rogers, James T., Jr. 5. Homeomorphism groups of weak solenoidal spaces. Proc. Amer. Math. Soc. 28 (1971) 242-246. (with Rogers, James T., Jr.) 6. Homogeneous inverse limit spaces with nonregular covering maps as bonding maps. Proc. Amer. Math. Soc. 29 (1971) 417-420.

Kelley, John E. 02 • 1. Peanian characterizations of E2 and S2. Proc. Amer. Math. Soc. 12 (1961) 907-916. 2. Compactness in the space of Radon measures. J. Functional Analysis 5 (1970) 259-298.

Morava, Jack J. 03 • 1. Fretholm maps ant Gysen homomorphisms. Global Analysis. (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) 135-136. American Mathematical Society, Providence, Rhode Island, 1970.

Grilliot, T. J. 03 • 1. Selection functions for recursive functionals. Notre Dame J. For- mal Logic 10 (1969) 225-234.

Boals, Alfred 03 • 216 CHAPTER 9

1. on-manifold factors of Euclidean spaces. Fund. Math. 68 (1970) 159-177.

Lubben, R. G. 01 • 1. Concerning limiting sets in abstract spaces. Trans. Amer. Math. Soc. 30 (1928) 668-685. 2. The double elliptic case of the Lie-Rieman-Helmholtz-Hilbert problem of the fountations of geometry. Fund. Math. 11 (1928) 35-95. 3. Separation theorems with applications to questions concerning approachability and plane continua. Trans. Amer. Math. Soc. 31 (1929) 503-52. 4. Concerning limiting sets in abstract spaces. III. Trans. Amer. Math. Soc. 43 (1938) 482-493. 5. Separabilities of arbitrary orders and related properties. Bull. Amer. Math. Soc. 46 (1940) 913-919. 6. Concerning the decomposition ant amalgamation of pts., upper semicontinuous collections, and topological extensions. Trans. Amer. Math. Soc. 49 (1941) 410-466. ACADEMIC DESCENDANTS AND PUBLICATIONS 217 Gordon T. Whyburn and his Mathematical Descendandts

01 Whyburn, Gordon T. 02 Harry, C. H., Johns Hopkins University, 1932 02 Aitchison, Barbara, Johns Hopkins University, 1933 02 Wheeler, Charles, III, Johns Hopkins University, 1933 02 Schweigart G. E., Johns Hopkins University, 1934 02 Wardwell , James F ., Johns Hopkins University , 1935 02 Hall, Dick Wick, University of Virginia, 1938 03 Haywoot, Stuart T., University of Maryland, 1950 03 Boyer, Jean Marie, University of Marylant, 1951 03 Siry, Joseph W., University of Maryland, 1953 03 Rector , Robert W., University of Maryland , 1956 03 Strohl , George R., University of Maryland , 1956 02 Wallace, A. D., University of Virginia, 1939 03 Saalfrank, Pennsylvania University, 1948 03 Butcher, George, Pennsylvania University, 1951 03 Conner, William, Tulane University, 1951 03 Keesee, John, Tulane University, 1951 04 Blakemore, Carol, University of Arkansas, 1968 04 BraSher, Rus, University of Arkansas, 1968 04 Starling, Greg, University of Arkansas, 1969 03 Strother, W. L., Tulane University, 1952 04 Day, Jane Maxwell, University of Florida, 1964 03 Cohen, Haskell, Tulane University, 1953 04 DeVun, Esmond E., University of Massachusetts, 1969 03 Gordon, William, Tulane University, 1953 03 Koch, Robert J., Tulane University, 1953 04 Rothman, N. J., Louisiana State University, 1958 05 Bergman , John G., University of Illinois , 1968 05 Schuh, Mark, University of Illinois, 1971 04 Hunter , R. P., Louisiana State Unlversity, 1958 05 Selden , John , Jr ., University of Georgia , 1963 06 Madison, Bernard, University of Kentucky, 1967 06 Stepp , James W., University of Kentucky, 1968 06 Kinch, L., University of Kentucky, 1969 05 Bastita, Jullo, University of Georgia, 1963 05 Stadtlander , David , Pennsylvania State Univ ., 1966 04 Brown, Dennison R., Louisiana State University, 1963 218 CHAPTER 9

05 Hildebrantt , J., University of Tennessee , 1965 05 Lawson, Jimmie, University of Tennessee, 1967 06 Lea, Jim, Louisiana State University, 1971 05 Farley, Reuben, University of Tennessee, 1968 05 Hinman, Betty, University of Houston, 1971 05 Lau , Yiu-Wa tAugust), University of Houston , 1971 04 Friedberg, Michael, Louisiana State University, 1964 04 Clark, C. E., Louisiana State University, 1966 05 Collins , Michael F., University of Missouri , 1970 04 Carruth, J. H., University of Missouri, 1966 05 Mislove, Michael W., University of Tennessee, 1969 04 Eberhart , C . A ., Louisiana State University , 1966 04 Williams, W. W., Louisiana State University, 1969 04 McCharen, J. D., Louisiana State University, 1969 04 L’Heureux, J. H., Louisiana State University, 1969 03 Ward, L. E., Tulane University, 1953 04 Smithson, Raymond E., Universty of Oregon, 1962 04 Harris, J. K., Univerity of Oregon, 1962 04 Lee, Yu-Lee, University of Oregon, 1969 05 Propes, Richard, Kansas State University, 1969 05 Yeung, Henderson C. H., Kansas State Univ., 1971 04 Dimitroﬀ, G. E., University of Oregon, 1964 04 Knight, Virginia Walsh, University of Oregon, 1967 04 Ferguson, E. N., University of Oregon, 1967 04 Mohler, L. K., University of Oregon, 1968 04 Tucker, L. D., University of Oregon, 1969 03 Anderson, Lee W., Tulane University, 1956 04 Stralka, Albert R., Pennsylvania State Univ., 1967 03 Faucett, William, Tulane University, 1955 03 Lin, You-Feng, Florida University, 1964 03 Borrego, J. T., Jr., Florida University, 1966 04 Sheldon, William L., University of Massachusetts, 1970 03 Lin, S. Y., Florida University, 1966 03 Choe, Tae Ho, Florida University, 1967 03 Sigmon, K. N., Florida University, 1967 03 Shershin, Anthony, Florida University, 1968 03 Chae, Y., Florida University, 03 Khuri, A., Florida University, 03 Robbie, D. A., Florida University, 02 Kelley, John L., University of Virginia, 1940 03 Fell, J. Michael Gardner, Univ. of Calif., Berkeley, 1951 03 Koosis, Paul Jacob, Univ. of Calif., Berkeley, 1954 ACADEMIC DESCENDANTS AND PUBLICATIONS 219

03 Prosser, Reese Trego, Univ. of Calif., Berkeley, 1955 03 Namioka, Isaac, Univ. of Calif., Berkeley, 1956 03 Bear, Herbert S., Jr., Univ. of Calif., Berkeley, 1957 03 Kallin, Eva Marianne, Univ. of Calif., Berkeley, 1963 03 Singh, Vashishta Narayan, Univ. of Calif., Berkeley, 1969 02 White, P. A., University of Virginia, 1942 02 Clark, C. L., University of Virginia, 1944 03 Hoggatt, Verner E., Jr., Oregon State University, 1955 02 Floyd, E. E., University of Virginia, 1948 02 Fort, M. K., University of Virginia, 1948 03 Lyttle, R. A., University of Georgia, 1955 03 Andrews, J. J., University of Georgia, 1957 04 Rubin, Leonard, Florida State University, 1965 04 Husch, Lawrence S., Florida State University, 1967 04 Tindell, Ralph S., Florida State University, 1967 03 Boswell, R. D., University of Georgia, 1957 03 Segal, Jack, University of Georgia, 1960 04 Clapp, Michael Howard, Univ. of Washington, 1968 04 Farmer, Frank Davis, Univ. of Washington, 1970 02 Kasriel, Robert H., University of Virginia, 1953 03 Cain, George L., Georgia Institute of Technology, 1965 04 Wertheimer, Stanley J., Georgia Inst. Tech., 1970 xxxxx 03 Fuller , Richard V ., Georgia Inst . of Technology , 1967 03 Brown , David L ., Georgia Institute of Technology , 1970 03 Wertheimer, Stanley J., Georgia Inst. of Tech., 1970 xxxx (with G. L. Cain) 02 Plunkett , R. L., University of Virginia, 1953 03 Morris, Joseph Richard, University of Alabama, 1969 02 Williams , Robert F., University of Virginia , 1954 03 Simon, Carl, Northwestern University, 1970 03 Gibbons, Joel, Northwestern University, 1970 02 Malbon, W. 1., University of Virginia, 1955 02 Jollensten, R. W., University of Virginia, 1956 02 McDougle , P . E ., University of Virginia , 1958 02 Duda, Edwin, University of Virginia, 1961 03 Keesling , James E ., University of Florida , 1968 03 Haynsworth, William Hugh, University of Florida, 1968 03 Smith, Jack Warren, University of Florida, 1969 02 Williams, G. K., University of Virginia, 1964 02 Duke , R. A., University of Virginia , 1965 02 Dickman , R. F ., University of Virginia , 1966 02 Stone, E. A., University of Virginia, 1966 02 Garcia-Maynez , A . C ., University of Virginia , 1968 220 CHAPTER 9

02 McMillan, Evelyn, University of Virginia, 1968 04 Friedberg, Michael 05 Karvellas, Paul, University of Houston, 1972 ACADEMIC DESCENDANTS AND PUBLICATIONS 221 Publications of Gordon T.Whyburn and his Mathematical Descendants

Whyburn, Gordon T. 02 • 1. Concerning certain types of continuous curves. Proc. Nat. Acad. Sci. 12 (1926 ) 7 61-7 67 . 2. Two-way continuous curves. Bull. Amer. Math. Soc. 32 (1926) 659-662 . 3. Concerning connected and regular point sets. Bull. Amer. Math. Soc. (1927) 685-689. 4. Concerning continua in the plane. Trans. Amer. Math. Soc. 29 (1927) 369-400. 5. Concerning point sets which can be made connected by the addition of a simple continuous arc. Trans. Amer. Math. Soc. 29 (1927) 746-754. 6. Concerning the disconnection of continua by the omission of pairs of their points. Fund. Math. 10 (1927) 180-185. 7. Concerning the open subsets of a plane continuous curve. Proc. Nat. Acad. Sci. 13 (1927) 650-657. 8. Cyclicly connected continuous curves. Proc. Nat. Acad. Sci. 13 (1927) 31-38 . 9. Some properties of continuous curves. Bull. Amer. Math. Soc. 33 (1927) 305-308 . 10. The most general closed point set over which a continuous function may be deﬁned by certain properties. Bull. Amer. Math. Soc. 33 (1927) 185-188. 11. Concerning accessibility in the plane and regular accessibility in n dimensions. Bull. Amer. Math. Soc. 34 (1928) 504-510. 12. Concerning Menger regular curves. Fund. Math. 12 (1928) 264- 294. 13. Concerning plane closed point sets which are accessible from certain subsets of their complements. Proc. Nat. Acad. Sci. 14 (1928) 657-666. 14. Concerning the complementary domains of continua. Ann. of Math. 29 (1928) 399-411. 15. Concerning the cut points of continua. Trans. Amer. Math. Soc. 39 (1928) 597-609. 222 CHAPTER 9

16. Concerning the structure of a continuous curve. Amer. J. Math. 50 (1928) 167-194. 17. On a problem of W L. Ayres. Fund. Math. 11 (1928) 298-301. 18. On certain accessible points of plane continua. Montash. Math. Phys. 35 (1928) 289-304. 19. On continuous curves in n-dimensions. Bull. Amer. Math. Soc. 34 (1928) 349-360. (with Ayres, W. L.) 20. The behavior of 2-phenyl semicarbazones upon oxidation. J. Amer. Chem. Soc. 50 (1928) 905-912. (with Bailey, J. R.) 21. A generalized notion of accessibility. Fund. Math. 14 (1929) 311-326. 22. Concerning collections of cuttings of connected point sets. Bull. Amer. Math. Soc. 35 (1929) 87-104. 23. Concerning points of continuous curves deﬁned by certain im kleinen properties. Math. Ann. 102 1929) 313-336. 24. Continuous curves and arc-sums. Fund. Math. 14 (1929) 103- 106. 25. Concerning irreducible cuttings of continua. Fund. Math. 13 (1929) 42-57. 26. Local separating points of continua. Montash. Math. Phys. 36 (1929) 305-314. 27. On regular points of continua and regular curves of at most order n. Bull. Amer. Math. Soc. 35 (1929) 218-224. 28. On simple closed curves. Bull. Internat. Acad. Polon. Sci. Lett. Cl. Sci. Math. Nat. Ser. A. 1929, No. 6A, 280-298. 29. A continuum every subcontinuum of which separates the plane. Amer. J. Math. 52 (1930) 319-330. 30. Cut points of connected sets and of continua. Trans. Amer. Math. Soc. 32 (1930) 147-154. 31. On the set of all cut points of a continuous curve. Fund. Math. 15 (1930) 185-194. 32. On the structure of connected and connected im kleinen point sets. Trans. Amer. Math. Soc. 32 (1930) 926-943. 33. ptntAllv regular point sets Fund. Math. 16 (1930) 160-172. 34. Sur l’accessibilite des continus plans. Fund. Math. 15 (1930) 322-323. ACADEMIC DESCENDANTS AND PUBLICATIONS 223

35. Sur les elements cycliques et leurs applications. Fund. Math. 16 (1930) 305-331. (with Kuratowski, C.) 36. Sur les separations irreducibles. Fund. Math. 16 (1930) 77-80. 37. The rationality of certain continuous curves. Bull. Amer. Math. Soc. 36 (1930) 522-524. 38. A junction property of locally connected sets. Amer. J. Math. 53 (1931) 753-756. 39. Concerning addition of regular curves. Monatsh. Math. Phys. 38 (1931) 1-4. 40. Concerning continuous images of the interval. Amer. J. Math. 53 (1931) 670-674. 41. Concerning hereditarily locally connected continua. Amer. J. Math. 53 (1931) 374-384. 42. Concerning the proposition that every closed compact and totally disconnected set of points is a subset of an arc. Fund. Math. 18 (1931) 47-60. 43. Concerning the subsets of regular curves. Monatsh. Math. Phys. 38 (1931) 85-88. 44. Continuous curves without local separating points. Amer. J. Math. 53 (1931) 163-166. 45. Non-separated cuttings of connected point sets. Trans. Amer. Math. Soc. 33 (1931) 444-454. 46. On the divisibility of locally connected spaces. Bull. Amer. Math. Soc. 37 (1931) 734-736. 47. On the cyclic connectivity theorem. Bull. Amer. Math. Soc. 37 (1931) 429-433. 48. The cyclic and higher connectivity of locally connected spaces. Amer. J. Math. 53 (1931) 427-442. 49. A certain transformation on metric spaces. Amer. J. Math. 54 (1932) 367-375. 50. A note on spaces which have the S property. Amer. J. Math. 54 (1932) 536-538. l 51. On the decomposability of closed sets into a countable number of simple sets of various types. Amer. J. Math. 54 (1932) 169-175. 52. On the construction of simple arcs. Amer. J. Math. 54 (1932) 518-524. 224 CHAPTER 9

53. Characterization of certain curves by continuous functions de- ﬁned upon them. Amer. J. Math. 55 (1933) 131-134. 54. Concerning S-regions in locally connected continua. Fund. Math. 20 (1933) 131-139. 55. Decompositions of continua by means of local separating points. Amer. J. Math. 55 (1933) 437-457. 56. On the existence of totally imperfect and punctiform connected subsets in a given continuum. Amer. J. Math. 55 (1933) 146- 152. 57. Sets of local separating points of a continuum. Bull. Amer. Math. Soc. 39 (1933) 97-100. 58. Concerning maximal sets. Bull. Amer. Math. Soc. 40 (1934) 159164. 59. Cyclic elements of higher orders. Amer. J. Math. 56 (1934) 133-146. 60. Non-alternating transformations. Amer. J. Math. 56 (1934) 294-302. 61. A decomposition theorem for closed sets. Bull. Amer. Math. Soc. 41 (1935) 95-96. 62. Concerning continua of ﬁnite degree and local separating points. Amer. J. Math. 57 (1935) 11-16. 63. Generalized perfect sets. Duke Math. J. 1 (1935) 35-38. 64. On sequences and limiting sets. Fund. Math. 25 (1935) 408-426. 65. Regular convergence and monotone transformations. Amer. J. Math. 57 (1935) 902-906. 66. Arc preserving transformations. Amer. J. Math. 58 (1936) 305312. 67. Completely alternating transformations. Fund. Math. 27 (1936) 140-146. 68. Concerning rationality bases for curves. Erebnisse eines, Math. Kolloq. 7 (1936) 58-60. 69. On continua of condensation. Amer. J. Math. 58 (1936) 705- 708. 70. On the structure of continua. Bull. Amer. Math. Soc. 42 (1936) 49-73. 71. Semi-closed sets and collections. Duke Math. J. 2 (1936) 685- 690. ACADEMIC DESCENDANTS AND PUBLICATIONS 225

72. Interior transformations on compact sets. Duke Math. J. 3 (1937) 370-381. 73. A theorem on interior transformations. Bull. Amer. Math. Soc. 44 (1938) 414-416. 74. Interior surface transformations. Duke Math. J. 4 (1938) 626- 634. 75. Interior transformations on certain curves. Duke Math. J. 4 (1938) 607-612. 76. Interior transformations on surfaces. Amer. J. Math. 60 (1938) 477-490. 77. The mapping of Betti groups under interior transformations. Duke Math. J. 4 (1938) 1-8. 78. Non-alternating interior retracting transformations. Ann. of Math. 40 (1939) 914-921. MRl,45. 79. On irreducibility of transformations. Amer. J. Math. 61 (1939) 820-822. MRl,45. 80. Semi-locally connected sets. Amer. J. Math. 61 (1939) 733-749. 81. The existence of certain transformations. Duke Math. J. 5 (1939) 647-655. MRl,30. 82. A relation between non-alternating and interior transformations. Bull. Amer. Math. Soc. 46 (1940) 320-321, MRl,319. 83. (with D. W. Hall), Arc- and tree-preserving transformations. Trans. Amer. Math. Soc. 48 (1940) 63-71. MRl,319. 84. Analytic topology. Amer. Math. Soc. Colloq. Publ., Vol. 28, Amer. Math. Soc., Providence, R. I., 1942; rev. ed., 1955. MR4,86. 85. On the interiority of real functions. Bull. Amer. Math. Soc. 48 (1942) 942-945. MR4,224. 86. What is a curve? Amer. Math. Monthly 49 (1942) 493-497; Studies in Modern Topology, Prentice-Hall, Englewood Cliﬀs, N.J., 1968 pp. 23-38. MR4,89; MR38,895. 87. Homotopy reductions of mappings into the circle. Duke Math. J. 11 (1944) 35-42. MR5,213. 88. Interior mappings into the circle. Duke Math. J. 11 (1944) 431- 434. MR6,164. 89. Topological analog of the Weierstrass double series theorem. Bull. Amer. Math. Soc. 50 (1944) 242-245. MR5,274. 226 CHAPTER 9

90. Boundary alternation of monotone mappings. Duke Math. J. 12 (1945) 663-667. MR7,336. 91. Coherent and saturated collections. Trans. Amer. Math. Soc. 57 (1945) 287-298. MR6,182. 92. Extensions of plane continua mappings. Amer. J. Math. 67 (1945) 505-520. MR7,136. 93. Uniqueness of the inverse of a transformation. Duke Math. J. 12 (1945) 317-323. MR7,36. 94. On monotone retractability into simple arcs. Bull. Amer. Math. Soc. 52 (1946) 109-112. MR7,468. 95. On locally simple curves. Bull. Amer. Math. Soc. 53 (1947) 986992. MR9,196. 96. On n-arc connectedness. Trans. Amer. Math. Soc. 63 (1948) 452-456. MR10,138. 97. Sequence approximations to interior mappings. Ann. Soc. Polon. Math. 21 (1948) 147-152. MR10,261. 98. Continuous decompositions. Amer. Math. Monthly 71 (1949) 218-226. MR10,317. 99. Modern development of mathematics. Va. J. Science 1 (1950) 93-102. 100. Open and closed mappings. Duke Math. J. 17 (1950) 69-74. MR11,194. 101. Open mappings on locally compact spaces. Mem. Amer. Math. Soc. No. 1, (1950) 24 pp. MR13,764. 102. An open mapping approach to Hurwitz’s theorem. Trans. Amer. Math. Soc. 71 (1951) 113-119. MRl3,149. 103. On k-fold irreducibility of mappings. Amer. J. Math. 74 (1952) 910-912. MR14,305. 104. On quasi-compact mappings. Duke Math. J. 19 (1952) 445-446. MR14,192. 105. A uniﬁed space for mappings. Trans. Amer. Math. Soc. 74 (1953) 344-350. MR14,669. 106. Introductory topological analysis. Lectures on Functions of a Complex Variable, Univ. of Michigan Press, Ann Arbor, Michi- gan, 1955, pp. 1-14. MR16,1140. 107. Relative quasi-compactness of mappings. Proc. Nat. Acad. Sci. 41 (1955) 974-978. MR19,568. ACADEMIC DESCENDANTS AND PUBLICATIONS 227

108. Set-theoretic topology-present and future, and Structure of continua. Summary of Lectures and Seminars, Summer Institute on Set Theoretic Topology. 1955, pp. 6. 67-68. 109. Mappings on inverse sets. Duke Math. J. 23 (1956) 237-240. MR20,4822. 110. Topological analysis. Bull. Amer. Math. Soc. 62 (1956) 204- 218. MR17,1229. 111. Dimension and non-density preservation of mappings. Paciﬁc J. Math. 7 (1957) 1243-1249. MR20,1299. 112. Quasi-open mappings. Rev. Math. Pures Appl. 2 (1957) 47-52. MR20,1300. 113. Sense and orientation on the disk. Amer. Math. Monthly 64 (1957) No. 8, part II, 103-106. MR20,4816. 114. Uniform convergence for monotone mappings. Proc. Nat. Acad. Sci. 43 (1957) 992-998. MR20,280. 115. On convergence of mappings. Colloq. Math. 6 (1958) 311-318. MR20,7256. 116. On the invariance of openness. Proc. Nat. Acad. Sci. 44 (1958) 464-466. MR21,2216. 117. Topological analysis. Princeton Math. Series, No. 23, Prince- ton Univ. Press, Princeton, N.J., 1958; 2nd rev. ed., 1964. MR20,6081; MR29,2758. 118. Topological characterization of the Sierpinski curve. Fund. Math. 45 (1958) 320-324. MR20,6077. 119. Compactness of certain mappings. Amer. J. Math. 81 (1959) 306314. MR22,1881. 120. Mapping norms. Proc. Nat. Acad. Sci. 45 (1959) 1431-1436. MR21,6578. 121. Convergence in norm. Proc. Nat. Acad. Sci. 46 (1960) 16141617. MR22,8476. 122. Non-obstructing sets and related mappings. Proc. Nat. Acad. Sci. 46 (1960) 1244-1247. MR23,A624. 123. Norms for open mappings. J. London Math. Soc. 35 (1960) 302-309. MR23,1825. 124. Compressibility and uniform convergence. Proc. Nat. Acad. Sci. (1961) 1843-1847. MR24,A2372. 228 CHAPTER 9

125. Open mappings on 2-dimensional manifolds. J. Math. Mech. 10 (1961) 181-197. MR27,2967. 126. A measure distortion mapping. Proc. Nat. Acad. Sci. 48 (1962) 19221924. MR26,1857. 127. Decomposition spaces. Topology of 3-manifolds and Related Topics (Proc. of Georgia Inst., 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 2-4. MR25,4499. 128. Developments in topological analysis. Fund. Math. 50 (1961/62) 305-318. MR29,3651. 129. Monotoneity of limit mappings. Duke Math. J. 29 (1962) 465- 470. MR26,6935. 130. The Cauchy inequality in topological analysis. Proc. Nat. Acad. Sci. 48 (1962) 1335-1336. MR32,4281. 131. Generic and related mappings. Bull. Amer. Math. Soc. 69 (1963) 757-761. MR28,1588. 132. Simplification of mappings. Proc. Nat. Acad. Sci. U.S.A 50 (1963) 431-435. MR27,4210. 133. On compactness of mappings. Proc. Nat. Acad. Sci. 52 (1964) 1426-1431. MR31,722. 134. Continuity of multifunctions. Proc. Nat. Acad. Sci. U.S.A 54 (1965) 1494-1501. M32,6423. 135. Directed families of sets and closedness of functions. Proc. Nat. Acad. Sci. U.S.A 54 (1965) 688-692. MR32,435. 136. Compactification of mappings. Math. Ann. 166 (1966) 168-174. MR34,791. 137. Connectivity of peripherally continuous functions. Proc. Nat. Acad. Sci. 55 (1966) 1040-1041. MR33,4902. 138. Functions and multifunctions, University of Virginia Lecture Notes, 1966/67. (assisted by John H. V. Hunt) 139. General topology and mapping theory, University of Virginia Lecture Notes. (assisted by R. F. Dickman, 1964/65; revised in 1966/67 with assistance of John H. V. Hunt, A. C. GarciaMaynes and Richard Sarchet) 140. Loosely closed sets and partially continuous functions. Mich. Math. J. 14 (1967) 193-205. MR34,8387. 141. Quasi-closed sets and fixed points. Proc. Nat. Acad. Sci. 57 (1967) 201-205. MR35,1006. ACADEMIC DESCENDANTS AND PUBLICATIONS 229

142. Cut points in general topological spaces. Proc. Nat. Acad. Sci. 61 (1968) 380-387. MR39,3463. 143. Introductory topology, University of Virginia Lecture Notes (manuscript). (with the assistance of W. C. Chewning and Margaret Moody) 144. Retracting multifunctions. Proc. Nat. Acad. Sci. 59 (1968) 343-348. MR37,3543. 145. Inward motions in connected spaces. Proc. Nat. Acad. Sci. 63 (1969) 271-274. 146. Accessibility spaces. Proc. Amer. Math. Soc. 24 (1970) 181- 185. MR40,1973. 147. Locally cohesive spaces. Proc. Kanpur Topological Conference, 1968. 148. Functional movements in dendritic structures. Proc. Kanpur Topological Conference, 1968. 149. Dynamic topology. Amer. Math. Monthly 77 (1970) 556-570.

Atchison, Barbara 02 • 1. Concerning regular accessibility. Fund. Math. 20 (1933) 116- 125. 2. On the mapping of locally connected continua into simple arcs. Comptes Rendus des Seances de la Societe des Sciences et des Lettres de Varsovie, Classe III, 27 (1934) 130-146.

Schweigert, G. E. 02 • 1. The analysis of certain curves by means of derived local separating points. Amer. J. Math. 58 (1936) 329-335. 2. A note on the limit of orbits. Bull. Amer. Math. Soc. 46 (1940) 963-969. 3. Minimal A-sets, inﬁnite orbits, and ﬁxed elements. ibid. 49 (1943) 754-758. 4. Fixed elements and periodic types for homeomorphisms on s. l. c. continua. Proc. Nat. Acad. Sci. 29 (1943) 52-54. 5. Fixed elements and periodic types for homeomorphisms on s. l. c. continua. Amer. J. Math. 66 (1944) 229-244. On the group of homeomorphisms of an arc. Ann. of Math. (2) 62 (1955) 237-253. (with Nathan J. Fine.) 230 CHAPTER 9

Wardwell, J. F. 02 • 1. Continuous transformations preserving all topological properties. Amer. J. Math. 58 (1936) 709-726.

Hall, D. W. 02 • 1. An example in the theory of pointwise periodic homeomorphisms. Bull. Amer. Math. Soc. 45 (1939) 882-885. 2. On a decomposition of true cyclic elements. Trans. Amer. Math. Soc. 47 (1940) 305-321. 3. A partial solution of a problem of J. R. Kline. Duke Math. J. 9 (1942) 893-901. 4. A note on primitive skew curves. Bull. Amer. Math. Soc. 49 (1943) 935-936. 5. On rotation groups of plane continuous curves under pointwise periodic homeomorphisms. ibid. 50 (1944) 715-718. 6. A note on Peano spaces. Proc. Amer. Math. Soc. 1 (1950) 231232. (with Boyer, Jean M.) 7. Periodic types of transformations. Duke Math. J. 8 (1941) 625- 630. (with Kelley, J. L.) 8. Coloring six-rings. Trans. Amer. Math. Soc. 64 (1948) 184-191. (with Lewis, J. C.) 9. Strongly arcwise connected spaces. Amer. J. Math. 63 (1941) 554-562. (with Puckett, W. T.,Jr.) 10. Conditions for the continuity of arc-preserving transformations. Bull. Amer. Math. Soc. 47 (1941) 468-475. 11. The chromatic polynomial of the truncated icosahedron. Proc. Amer. Math. Soc. 16 (1965) 620-628. (with Siry, J. W.; Van- derslice, B. R.) 12. Arc and tree-preserving transformations. Trans. Amer. Math. Soc. 48 (1940) 63-71. (with Whyburn, G. T.) 13. Elementary topology. John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd. London, 1955. xii+303 pp. (with Spencer, Guilford, II.) 14. Comments on the cores of certain classes of spaces. Ann. of Math. (2) 48 (1947) 710-716. (with Youngs, J. W. T.)

Boyer, Jean Marie 03 • ACADEMIC DESCENDANTS AND PUBLICATIONS 231

1. A note on Peano spaces. Proc. Amer. Math. Soc. 1 (1950) 231-232. (with Hall, D. W.)

Siry, Joseph W. 03 • 1. Satellite lauching vehicle trajectories. Proc. Sympos. Appl. Math. Vol. 9, pp. 75-144. American Mathematical Society, Providence, R. I., 1959. 2. The chromatic polynomial of the truncated icosahedron. Proc. Amer. Math. Soc. 16 (1965) 620-628. (with Hall, D. W.; Van- derslice, B. R.)

Strohl, G. Ralph 03 • 1. Peano spaces which are either strongly cyclic or two-cyclic. Trans. Amer. Math. Soc. 86 (1957) 297-308.

Kelley, John L. 02 • 1. A metric connected with property S. Amer. J. Math. 61 (1939) 764-768 . 2. Fixed sets under homeomorphisms. Duke Math. J. 5 (1939) 535-537. 3. A decomposition of compact continua and related theorems on fixed sets under continuous transformations. Proc. Nat. Acad. Sci. 26 (1940) 192-194 . 4. Hyperspaces of a continuum. Trans. Amer. Math. Soc. 52 (1942) 22-36. 5. Simple links and fixed sets under continuous mappings. Amer. J. Math. 69 (1947) 348-356. 6. Convergence in topology. Duke Math. J. 17 (195G) 277-283. 7. The Tychonoff product theorem implies the axiom of choice. Fund. Math. 37 (1950) 75-76. 8. Note on a theorem of Krein and Milman. J. Osaka Inst. Sci. Tech. Part I. 3 (1951) 1-2. 9. Commutative operator algebras . Proc. Nat. Acad. Sci. 38 (1952) 598-605 . 10. Banach spaces with the extension property. Trans. Amer. Math. Soc. 72 (1952) 323-326. 232 CHAPTER 9

11. General topology. D. Van Nostrand Company, Inc., Toronto- New YorkLondon, 1955, xiv+298 pp. (A. H. Stone) 16-1136. 12. On mappings of plane sets. Colloq. Math. 6 (158) 153-154. 13. Hypercomplete linear topological spaces. Bull. Soc. Math. Belg. 10 (1958) 2-3. 14. Averaging operators on C (X). Illinois J. Math. 2 (1958) 214- 223. 15. Hypercomplete linear topological spaces. Mich. Math. J. 5 (1958) 235-246 . 16. Measures on Boolean algebras. Paciﬁc J. Math. 9 (1959) 11651177 . 17. Introduction to modern algebra. The University Series in Un- dergraduate Mathematics. D. Van Nostrand Co., Inc. Princeton, N. J., Toronto-London-New York, 1960. x+338 pp. 18. Descriptions of Cech cohomology. General Topology and its Re- lations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) pp. 235-237. Academic Press, New York; Publ. House Czech. Acad. Sci., Progue, 1962. 19. Duality for compact groups. Proc. Nat. Acad. Sci. 49 (1963) 457-458. 20. Decomposition and representation theorems in measure theory. Math. Ann. 163 (1966) 89-94. 21. General Topology. Editorial Universitaria de Buenos Aires, Buenos Aires, 1962, 336 pp. 22. Characterizations of the space of continuous functions over a compact Hausorﬀ space. Trans. Amer. Math. Soc. 62 (1947) 499508. (with Arens, R. F.) 23. An algebra of unbounded operators. Proc. Nat. Acad. Sci. U.S.A 38 (1952) 592-598. (with Fell, J. M. G.) (with Hall, D. W.) 24. Periodic types of transformations. Duke Math. J. 8 (1941) 625- 630. (with Hall, D. W.) 25. Exterlod ballistics. The University of Denver Press, Denver, Colo., 1953, xii+834 pp. (1 plate). (with McShane, E. J.; Franklin, V.) 26. Invariant measures. Math. Ann. 148 (1962) 98-124. (with Morse A. P.) ACADEMIC DESCENDANTS AND PUBLICATIONS 233

27. Linear topological spaces. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbethue Paulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, J. J., 1963, xv+256 pp. (with Namioka, Issac) 28. Exact homomorphism sequences in homology theory. Ann. of Math. (2) 48 (1947) 682-709. (with Pitcher, Everett.) 29. The positive cone in Banach algebras. Trans. Amer. Math. Soc. 74 (1953) 44-55. (with Vaught, R. L.)

Fell J. M. G. 03 • 1. The measure ring for a cube of arbitrary dimensions. Pacific J. Math. 5 (1955) 513-517. 2. A note on abstract measure. ibid. 6 (1956) 43-45. 3. Representations of weakly closed algebras. Math. Ann. 133 (1957) 118-126. 4. C -algebras with smooth dual. Illinois J. Math. 4 (1960) 221- 230. 5. The dual spaces of C -algebras. Trans. Amer. Math. Soc. 94 (1960) 365-403. 6. Weak containment and induced representations of groups. Canad. J. Math. 14 (1962) 237-268. 7. A new proof that nilpotent groups are CCR. Proc. Amer. Math. Soc. 13 (1962) 93-99. 8. A Hausdorff topology for the closed subsets of a locally compact nonHausdorff space. Proc. Amer. Math. Soc. 13 (1962) 472- 476. 9. Weak containment and Kronecker products of group representations. Pacific J. Math. 13 (1963) 503-510. 10. Weak containment and induced representations of groups. II. Trans. Amer. Math. Soc. 110 (1964) 424-447. 11. The structure of algebras of operator fields. Acta. Math. 106 (1961) 233-280. 12. The dual spaces of Banach algebras. Trans. Amer. Math. Soc. 114 (1954) 227-250. 13. Non-unitary dual spaces of groups. Acta. Math. 114 (1965) 267-310. 234 CHAPTER 9

14. Algebras and fiber bundles. Pacific J. Math. 16 (1966) 497-503. 15. Comparisor. des mesures portees par unensemble convexe compact. Bull. Soc. Math. France 92 (1964) 435-445. (with Cartier, Pierre; Meyer, Paul-Andre.) 16. Conjugating representations and related results on semisimple Lie groups. Trans. Amer. Math. Soc. 127 (1967) 405-426. 17. An extension of Mackey’s method to algebraic bundles over finite groups. Amer. J. Math. 91 (1969) 203-238. 18. Separable representations of rings of operators. Ann. of Math. (2) 65 (1957) 241-249. (with Feldman, Jacob.) 19. An algebra of unbounded operatores. Proc. Nat. Acad. Sci. 38 (1952) 592-598. (with Kelley, J. L.) 20. On algebras whose factor algebras are Boolean. Pacific J. Math. 2 (1952) 297-318. (with Tarski, Alfred.) 21. Eine Bemerkungen uber vollsymmetrische Banachshe Algebren. Arch. Math. 12 (1961) 69-70. (with Thoma, Elmar.)

Koosis Paul 03 • 1. Note sur les fonctions moyenne-periodiques. Ann. Inst. Fourier, Grenoble 6 (1955-1956) 357-360. 2. One dimensional repeating curves in the non-degenerate case. Contributions to the theory of nonlinear oscillations, vol. 3, pp. 277-285. Annals of Mathematics Studies, no. 36. Princeton University Press, Princeton, J. J., 1956. 3. On functions ich are mean periodic on a half-line. Comm. Pure Appl. Math. 10 (1957) 133-149. 4. Interior compact spaces of functions on a half-line. Comm. Pure Appl. Math. 10 (1957) 583-615. 5. A completeness theorem. Portugal. Math. 15 (1956) 111-113. 6. Approximation of certain functions by exponentials on a half line. Proc. Amer. Math. Soc. 8 (1957) 428-435. 7. An irreducible unitary representation of a compact group is ﬁnite dimensional. Proc. Amer. Math. Soc. 8 (1957) 712-715. 8. Sur la non-totalite de certaines suites d’exponentielles sur des intervalles assez longs. Ann. Sci. Ecole Norm. Sup. (3) 75 (1958) 125-152. ACADEMIC DESCENDANTS AND PUBLICATIONS 235

9. Nouvelle demonstration des zeros d’une fonction de type expo- nentiel, bornee sur l’axe reel. Bull. Soc. Math. France 86 (1958) 27-40. 10. Proof of a theorem of the brothers Riesz. Studia Math. 17 (1958) 295-298. 11. Les sous-espaces interieurement compacts de 12 (0,). Seminaire P. Lelong, 1968/69, exp. 9, 4 pp. Faculte des Sciences de Paris, 1959. 12. Sur un theoreme remarquable de M. Malliavin. C. R. Acad. Sci. Paris 249 (1959) 352-354. 13. Sur la totalite des systemes d’ exponentielles imaginaires. C. R. Acad. Sci. Paris 250 (1960) 2102-2103. 14. Sur un theoreme de Paul Cohen. C. R. Acad. Sci. Paris 259 (1969) 1380-1382. 15. Sue l’approximation ponderee par des polynomes et par des sommes d’exponentielles imaginaires. Ann. Sci. Ecole Norm. Sup. (3) 81 (1964) 387-408. 16. On the spectral analysis of bounded functions. Paciﬁc J. Math. 16 (1966) 121-128. 17. Solution du probleme de Bernstein sur les entiers. C. R. Acad. Sci. Paris Ser. A-B 262 (1966) A1100-A1102. 18. L’approximation ponderee sur des progressions arithmetiques d’intervalles ou de points. C. R. Acad. Sci. Paris 26 (1965) 30223024. 19. Weighted polynomial approximation on arithmetic progressions of intervals or points. Acta. Math. 116 (1966) 223-277.

Prosser R. T. 03 • 1. On the consistency of quantum ﬁeld theory. Bull. Amer. Math. Soc. 69 (1963) 552-557. 2. Relativistic potential scattering. J. Math. Phys. 4 (1963) 1048- 1054. 3. On the ideal structure of operator algebras. Mem. Amer. Math. Soc. No. 45 (1963) ii+28 pp. 4. Segal’s quantization procedure. J. Math. Phys. 5 (1964) 701- 707. No. 45 (1963) ii+28 pp. 236 CHAPTER 9

5. Convergent perturbation expansions for certain wave operators. J. Math. Phys. 5 (1964) 708-713. 6. A new formulation of particle mechanics. Mem. Amer. Math. Soc. No. 61 (1966) 57 pp. 7. A multidimensional sampling theorem. J. Math. Anal. Appl. 16 (1966) 574-584. 8. The xxxx-entropy and xxxx-capacity of certain time-varying channels. J. Math. Anal. Appl. 16 (1966) 553-574. 9. On a similarity invariant for compact operators. Trans. Amer. Math. Soc. 134 (1968) 171-181. 10. Formal solutions of inverse scattering problems. J. Math. Phys. 10 (1969) 1819-1822. (with Root, W. L.) 11. Determinable classes of channels. J. Math. Mech. 16 (1966) 365-397. (with Root, W. L.) 12. The xxx-entropy and xxxx-capacity of certain time-invariant channels. J. Math. Anal. Appl. 21 (1968) 233-241.

Namioka, Issac 03 • 1. Partially ordered linear topological spaces. Mem. Amer. Math. Soc. No. 24 (1957) 50 pp. 2. A substitute for Lebesgue’s bounded convergence theorem. Proc. Amer. Math. Soc. 12 (1961) 713-716. 3. On certain onto maps. Canadian J. Math. 14 (1962) 461-466. 4. Maps of pairs in homotopy theorem. Proc. London Math. Soc. (3) 12 (1962) 725-738. 5. Folner’s conditions for amenable semi-groups. Math. Scand. 15 (1964) 18-28. 6. A duality in function spaces. Trans. Amer. Math. Soc. 115 (1965) 131-144. 7. On a recent theorem by H. Reiter. Proc. Amer. Math. Soc. 17 (1966) 1101-1102. 8. Neighborhoods of extreme points. Israel J. Math. 5 (1967) 145152. On certain actions of semi-groups on L-spaces. Stu- dia Math. 29 (1967) 63-77. 9. H-spaces with commutative homology rings. Ann. of Math. (2) 75 (1962) 449-451. (with Browder, William.) ACADEMIC DESCENDANTS AND PUBLICATIONS 237

10. Extreme invariant positive operators. Trans. Amer. Math. Soc. 137 (1969) 375-385. (with Converse, George and Phelps, R. R.) 11. Linear topological spaces. The University Series in Higher Math- ematics. D. Van Nostrand Co., Inc., Princeton, N. J., 1963 xv+256 pp. (with Kelley, J. L.) With the collaboration of W. J. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbethue Paulsen, G. Bailey Price, Wendy Robertson, W. R. Scott, Ken- nan T. Smith.

Bear, H. S., Jr. 03 • 1. Complex function algebras. Trans. Amer. Math. Soc. 90 (1959) 383-393. 2. A strong maximum modulus theorem for maximal function algebras. Trans. Amer. Math. Soc. 92 (1959) 464-469. 3. Some boundary properties of function algebras. Proc. Amer. Math. Soc. 11 (1960) 1-4. 4. The Silov boundary for a linear space of continuous functions. Amer. Math. Monthly 68 (1961) 483-485. 5. An abstract potential theory with continuous kernel. Paciﬁc J. Math. 14 (1964) 407-420. 6. A strict maximum theorem for one-part function spaces and algebras. Bull. Amer. Math. Soc. 70 (1964) 642-643. 7. A geometric characterization of Gleason parts. Proc. Amer. Math. Soc. 16 (1965) 407-412. 8. The integral representation of functions on parts. Illinois J. Math. 10 (1966) 49-55. 9. Continuous subparts for function spaces. Function Algebras (Proc. Internat. Sypos. on Function Algebras, Tulane Univ., 1965) pp. 292-299. Scott-Foresman, Chicago, Ill., 1966. 10. A general chaining lemma. Amer. Math. Monthly 76 (1969) 791-795. 11. A global integral representation for abstract harmonic functions. J. Math. Mech. 16 (1967) 639-653. (with Gleason, A. M.) 12. Integral kernel for one-part function spaces. Paciﬁc J. Math. 23 (1967) 209-215. (with Walsh, Bertram) 13. An intrinsic metric for parts. Proc. Amer. Math. Soc. 18 (1967) 812-817. (with Weiss, Max E.) 238 CHAPTER 9

Kallin, Eva-Marianne 03 • 1. A nonlocal function algebra. Proc. Nat. Acad. Sci. 49 (1963) 821-842. 2. Polynomial convexity: The three spheres problem. Proc. Conf. Complex Analysis, Minneapolis, 1964, 301-304. Springer, Berlin, 1965. 3. Fat polynomially convex sets. Function Algebras (Proc. In- ternat. Sympos. on Function Algebras, Tulane Univ., 1965) 149-152. Scott-Foresman, Chicago, Ill., 1966.

White Paul A. 02 • 1. r-regular convergence spaces. Amer. J. Math. 66 (1944) 69-96. 2. On r-regular convergence. Bull. Amer. Math. Soc. 50 (1944) 132-128. 3. Additive properties of compact sets. Duke Math. J. 11 (1944) 699-701. 4. Regular transformations. Duke Math. J. 12 (1945) 101-106. 5. New types of regular convergence. Duke Math. J. 12 (1945) 305-315. 6. On the equivalence between avoidability and co-local connectedness. Anais Acad. Brasil. Ci. 19 (1947) 143-151. 7. Regular transformations on generalized manifolds. Duke Math. J. 14 (1947) 769-775. 8. On a certain class of set theoretic properties. Ann. Mat. Pura Appl. (4) 31 (1950) 99-110. 9. On the union of wo generalized manifolds. Ann. Scoula Norm. Super. Pisa (3) 4 (1950) 231-243. 10. Regular convergence in terms of Cech cycles. Ann. of Math. (2) 55 (1952) 420-432. 11. Some characterizations of generalized manifolds with boundaries. Canadian J. Math. 4 (1952) 329-342. 12. Extensions of the Jordan-Brouwer separation theorem and its converse. Proc. Amer. Math. Soc. 3 (1952) 488-498. 13. Regular convergence of manifolds with boundary. Proc. Amer. Math. Soc. 4 (1953) 482-485. ACADEMIC DESCENDANTS AND PUBLICATIONS 239

14. Regular convergence. Bull. Amer. Math. Soc. 60 (1954) 431- 443. 15. The computation of eigenvalues and eigenvectors of a matrix. J. Soc. Indust. Appl. Math. 6 (1958) 393-437.

Hoggatt, Verner, E., Jr. 03 • 1. Fibonacci numbers and generalized binomial coeﬃcients. Fi- bonacci Quart. 5 (1967) 383-400. 2. A new angle on Pascal’s triangle. Fibonacci Quart. 61 (1968) No. 4, 221-234. 3. Generalized rabbits for generalized Fibonnacci numbers. Fi- bonacci Quart. 6 (1968) No. 3, 105-112. 4. Symbolic substitutions into Fibonnaci polynomials. Fibonacci Quart. 6 (1968) No. 5., 55-74. 5. Fibonacci matrices and lambda functions. Fibonacci Quart. (1963) No. 2, 47-52. 6. Fourth power Fibonacci identitites from Pascal’s triangle. Fi- bonacci Quart. 2 (1964) 261-266. 7. Hyperbolic trigonometry derived from the Poincare model. Amer. Math. Monthly 58 (1951) 469-474. (with Eves, Howard) 8. A power identity for second-order recurrent sequences. Fibonacci Quart. 4 (1966) 274-282. (with Lind, D. A.) 9. The heights of Fibonacci polynomials and an associated function. Fibonacci Quart. 5 (1967) 141-152. 10. Fibonacci and binomial properties of weighted compositions. J. Combinatorial Theory 4 (1968) 121-124. (with Lind, D. A.) 11. A primer for the Fibonacci numbers. VI. Fibonacci Quart. 5 (1967) 445-460.

Floyd, E. E. 02 • 1. A nonhomogeneous minimal set. Bull. Amer. Math. Soc. 55 (1949) 957-960. 2. The extension of homeomorphisms. Duke Math. J. 16 (1949) 225-235. 3. Some characterizations of interior maps. Ann. of Math. 51 (1950) 571-575. 240 CHAPTER 9

4. Coverings with connected intersections. Trans. Amer. Math. Soc. 69 (1950) 387-391. 5. On the extension of homeomorphisms on the interior of a two cell. Bull. Amer. Math. Soc. 52 (1946) 654-658. 6. A nonhomogeneous minimal set. Bull. Amer. Math. Soc. 55 (1949) 957-960. 7. The extension of homeomorphisms. Duke Math. J. 16 (1949) 225-235. 8. Some characterizations of interior maps. Ann. of Math. (2) 51 (1950) 571-575. 9. Some retraction properties of the orbit decomposition spaces of periodic maps. Amer. J. Math. 73 (1951) 363-367. 10. On related periodic maps. Amer. J. Math. 74 (1952) 547-554. 11. Examples of fixed point sets of periodic maps. Ann. of Math. (2) 55 (1952) 167-171. 12. On periodic maps and the Euler characteristics of associated spaces. Trans. Amer. Math. Soc. 72 (lg52) 138-147. 13. Orbit spaces of finite transformation groups. I. Duke Math. J. 20 (1953) 563-567. 14. Orbit spaces of finite transformation groups. II. Duke Math. J. 22 (1955) 33-38. 15. Boolean algebras with pathological order topologies. Pacific J. Math. 5 (1955) 687-689. 16. Real-valued mappings of spheres. Proc. Amer. Math. Soc. 6 (1955) 17. Examples of fixed point sets of periodic maps. II. Ann. of Math. (2) 64 (1956) 396-398. 18. Orbits of torus groups operating on manifolds. Ann. of Math. (2) 65 (1957) 505-512. 19. Fixed point sets of compact abelian Lie groups of transformations. Ann. of Math. (2) 66 (1957) 30-35. 20. Closed coverings in Cech homology theory. Trans. Amer. Math. Soc. 84 (1957) 319-337. 21. Some connections between cobordism and transformation groups. Proc. Internat. Congr. Mathematicians (Stockholm, 1962) pp. 462466. Inst. Mittag-Leffler, Djursholm, 1963. ACADEMIC DESCENDANTS AND PUBLICATIONS 241

22. Chain theories and their derived homology. Proc. Amer. Math. Soc. 19 (1968) 1115-1118. (with Burdick, R. O. and Conner, P. E.) 23. A characterization of generalized manifolds. Mich. Math. J. 6 (1959) 33-43. (with Conner Pierre E.) 24. On the construction of periodic maps without fixed points. Proc. Amer. Math. Soc. 10 (1959) 354-360. 25. Differentiable periodic maps. Bull. Amer. Math. Soc. 68 (1962) 76-86. 26. Fixed point free involutions and equivariant maps. II. Trans. Amer. Math. Soc. 105 (1962) 222-228. 27. A note on the action of S0(3). Proc. Amer. Math. Soc. 10 (1959) 616-620. 28. Fixed point free involutions and equivariant maps. Bull. Amer. Math. Soc. 66 (1960) 416-441. 29. Periodic maps which preserve complex structure. Bull. Amer. Math. Soc. 70 (1964) 574-579. 30. The SU-bordism theory. Bull. Amer. Math. Soc. 70 (1964) 670-675. 31. Differentiable periodic maps. Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Gottingen-Heidelberg, 1964. vii+148 pp. 32. Fibrixxxx within a cobordism class. Mich. Math. J. 12 (1965) 33-47. 33. Torsion in SU-bordism. Mem. Amer. Math. Soc. No. 60 (1966) 74 pp. 34. Maps of odd period. Ann. of Math. (2) 84 (1966) 132-156. 35. The relation of cobordism to K-theories. Springer-Verlag, BerlingNew York, 1966, v+112 pp. 36. Coverings with connected intersections. Trans. Amer. Math. Soc. 69 (1950) 387-391. (with Bing, R. H.) 37. A characterization theorem for monotone mappings. Proc. Amer. Math. Soc. 4 (1953) 828-830. (with Fort, M. K., Jr.) 38. A characterization of reflexivity by the lattice of closed subspaces. Proc. Amer. Math. Soc. 5 (1954) 655-661. (with Klee, V. L.) 242 CHAPTER 9

39. An action of a ﬁnite group on an n-cell without stationary points. Bull. Amer. Math. Soc. 65 (1959) 73-76. (with Richardson, R. W.)

Fort, M. K., Jr. 02 • 1. A specialization of Zorn’s lemma. Duke Math. J. 15 (1948) 763-765. 2. A unified theory of semi-continuity. ibid. 16 (1949) 237-246. 3. A note in equicontinuity. Bull. Amer. Math. Soc. 55 (1949) 1098-1100. 4. Essential and non essential fixed points. Amer. J. Math. 72 (1950) 316-322. 5. Open topological disks in the plane. J. Indian Math. Soc. (N.S.) 18 (1954) 23-26. 6. A proof that the group of all homeomorphisms of the plane onto itself is locally arcwise connected. Proc. Amer. Math. Soc. 1 (1950) 59-62. 7. A note on pointwise convergence. Proc. Amer. Math. Soc. 2 (1951) 34-35. 8. A characterization of plane light open mappings. Proc. Amer. Math. Soc. 2 (1951) 175-177. 9. Points of continuity of semi-continuous functions. Publ. Math. Debrecen 2 (1951) 100-102. 10. Some properties of continuous functions. Amer. Math. Monthly 59 (1952) 372-375. 11. A cylindrical curve with maximum length and maximum height. Quart. J. Math., Oxford Ser. (2) 4 (1953) 314-320. 12. A theorem about topological n-cells. Proc. Amer. Math. Soc. 5 (1954) 456-459. 13. Category theorems. Fund. Math. 42 (1955) 276-288. 14. The embedding of homeomorphisms in flows. Proc. Amer. Math. Soc. 6 (1955) 960-967. 15. Essential mappings. Amer. Math. Monthly 63 (1956) 238-241. 16. A geometric problem of Sherman Stein. Pacific J. Math. 6 (1956) 607-609. 17. Extensions of mappings into n-cubes. Proc. Amer. Math. Soc. 7 (1956) 539-542. ACADEMIC DESCENDANTS AND PUBLICATIONS 243

18. A note concerning a decomposition space defined by Bing. Ann. of Math. (2) 65 (1957) 501-504. 19. Research problem number 22. Math. Student 24 (1956) 189-191. 20. Mappings on Sl into one-dimensional spaces. Illinois J. Math. (1957) 505-508. 21. Images of plane continua. Amer. J. Math. 81 (1959) 541-546. 22. E -mappings of a disc onto a torus. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 7 (1959) 51-54. 23. Extension of the set on which mappings into S are homotopic. Fund. Math. 48 (1959/60) 271-276. 24. Neighborhood extensions of continuous selections. Proc. Amer. Math. Soc. 1 (1960) 682-685. 25. The complements of bounded, open, connected subsets of Eu- clidean space. (Russian summary) Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 9 (1961) 457-460. One-to-one mappings onto the Cantor set. J. Indian Math. Soc. (N.S.) 25 (1961) 103-107 (1962). 26. Sequences of homeomorphisms on the n-sphere. Proc. Amer. Math. Soc. 12 (1961) 361-363. 27. Homogeneity of infinite products of manifolds with boundary. Topology of 3-manifolds and reltaed topics. (Proc. The Univ. of Georgia Institute, 1961), pp. 42-43. Prentice-Hall, Inc. En- glewood Cliffs, N.J., 1962. 28. (Editor). Topology of 3-manifolds and related topics. Proceed- ings of the University of Georgia Institute, 1961. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. ix+256 pp. 29. Homogeneity of infinite products of manifolds with boundary. Pacific J. Math. 12 (1962) 879-884. 30. Continuous solutions of a functional equation. Ann. Polon. Math. 13 (1963) 205-211. 31. A wild sphere which can be pierced at each point by a straight line 1. segment. Proc. Amer. Math. Soc. 14 (1963) 994-995. 32. Homotopy for cellular set-valued functions. Proc. Amer. Math. Soc. 16 (1965) 455-459. (with Brahana, T. R.; Horstman, Walt G.) 33. Certain subgroups of the homotopy groups. Mich. Math. J. 4 (1957) 167-172. (with Curtis, M. L.) 244 CHAPTER 9

34. Homotopy groups of one-dimensional spaces. Proc. Amer. Math. Soc. 8 (1957) 577-579. 35. Singular homology of one-dimensional spaces. Proc. Amer. Math. Soc. (2) 69 (1959) 309-313. (with Curtis, M. L.) 36. The fundamental group of one-dimensional spaces. Proc. Amer. Math. Soc. 10 (1959) 140-148. 37. A characterization theorem for monotone mappings. Proc. Amer. Math. Soc. 4 (1953) 828-830. (with Floyd, E. E.) 38. Minimal coverings of pairs by triples. Paciﬁc J. Math. 8 (1958) 709-719. (with Hedlund, G. A.) 39. Sui teoremi di convergenza delle medie nei processi non stazionari. Ann. Mat. Pura Appl. (4) 56 (1961) 263-279. 40. Processi ergodici non stazionari e lo.o proprieta. Ann. Math. Pura. Appl. (4) (1962) 351-356. 41. Approximation of maps of inverse l mit spaces by induced maps. Fund. Math. 59 (1966) 323-329. (with McCord, M. C.) 42. Local connectedness of inverse li it spaces. Proc. Amer. Math. Soc. 10 (1959) 253-259. (with Segal, Jack.) 43. Minimal representations of the hyperspace of a continuum. Duke Math. J. 32 (1965) 129-137. 44. Convergence of series whose terms are deﬁned recursively. Amer. Math. Monthly 71 (1964) 994-998. (with Shuster, Seymour.) 45. A solution to the restricted ergocic problem in statistical mechanics. (Italian summary) Rend. Mat. Univ. Parama (2) 3 (1962) 77-7. (with Strocchi, Franco.)

Andrews James J. 03 • 1. A characterization of light open maps of Euclidean spaces into Euclidean spaces. Proc. Amer. Math. Soc. 9 (1958) 860-861. 2. A chainable continuum no two of whose nondegenerate subcontinua are homeomorphic. Proc. Amer. Math. Soc. 12 (1961) 333-334. 3. Embedding homotopy cells. Proc. Amer. Math. Soc. 12 (1961) 917. 4. N-space modulo an arc. Topology of 3-manifold and reltaed topics. (Proc. The Univ. of Georgia Institute, 1961) p. 44. PrenticeHall, Englewood Cliﬀs, N.J., 1962. ACADEMIC DESCENDANTS AND PUBLICATIONS 245

5. Knotted 2-spheres in the 4-sphere. Ann. of Math. (2) 70 (1959) 565-571. (with Curtis, M. L.) 6. N-space modulo an arc. Ann. of Math. (2) 75 (1962) 1-7. 7. Free groups and handlebodies. Proc. Amer. Math. Soc. 16 (1965) 192-195. 8. Extended Nielsen operations in free groups. Amer. Math. Monthly 73 (1966) 21-28. (with Curtis. M. L.) 9. The Minkowski units of ribbon knots. Proc. Amer. Math. Soc. 15 (1964) 856-864. (with Dristy, Forrest.) 10. The second homotopy group of spun l-spheres in 4-space. Bull. Amer. Math. Soc. 75 (1969) 169-171. (with Lomonaco, S. J.) 11. The second homotopy group of spun 2-spheres in r-space. Ann. of Math. (2) 90 (1969) 199-204. 12. Some spaces whose product with El is E4. Bull. Amer. Math. Soc. 71 (1965) 675-677. (with Rubin, Leonard.)

Rubin, Leonard 04 • 1. The product of an unusual decomposition space with a line is E4. Duke Math. J. 33 (1966) 323-329. 2. Some spaces whose product wit El is E4. Bull. Amer. Math. Soc. 71 (1965) 675-677. (with Andrews, J. J.) 3. Semigroups and clusters of indecomposability. Fund. Math. 56 (1964) 21-33. (with Dickman, R. F.; Kelley, J. L.; and Swingle. P. M.) 4. Another characterization of the n-sphere and related results. Pa- ciﬁc J. Math. 14 (1964) 871-878. (with Dickman, R. F. and Swingle, P. M.) 5. Irreducible continua and generalization of hereditarily unicoherent continua by means of membranes. J. Austral. Math. Soc. 5 (1965) 416-426.

Husch, L. S. 04 • 1. On regular neighborhoods of spheres. Bull. Amer. Math. Soc. 72 (1966) 879-881. 2. The monotone union property of manifolds. Trans. Amer. Math. Soc. 131 (1969) 345-355. 3. On collapsible ball pairs. Illinois J. Math. 12 (1968) 414-420. 246 CHAPTER 9

4. Mapping cylinders and the annulus conjecture. Bull. Amer. Math. Soc. 21 (1969) 506-508. 5. Finding a boundary for a 3-manifold. Proc. Amer. Math. Soc. 21 (1969) 64-68. 6. Desuspending collapses. Nederl. Acad.Wetensch. Proc. Ser. A 72 Indag. Math. 31 (1969) 285-291. 7. Unknotting in codimension one. Proc. Amer. Math. Soc. 23 (1969) 215-219. 8. On relative regular neighbourhoods. Proc. London Math. Soc. (3) 19 (1969) 577-585.

Tindell, Ralph A. 04 • 1. A counterexample on relative regular neighborhoods. Bull. Amer. Math. Soc. 72 (1966) 892-893. 2. Some wild imbeddings in codimension two. Proc. Amer. Math. Soc. 17 (1966) 711-716. 3. A mildly wild two-cell. Mich. Math. J. 13 (1966) 449-457. 4. Knotting tori in hyperplanes. Conference on the Topology of Manifolds ( Michigan State Univ., E. Lansing, Michigan, 1967) pp. 147153. Prindle, Weber, Schmidt, Boston, Mass., 1968. 5. Extending homeomorphisms of SP x S . Proc. Amer. Math. Soc. 22 (1969) 230-232. 6. Noncombinatorial triangulations and the Poincare conjecture. Proc. Amer. Math. Soc. 24 (1970) 60-62.

Segal, Jack 03 • 1. Hyperspaces of the inverse limit space. Proc. Amer. Math. Soc. 10 (1959) 706-709. 2. A ﬁxed point theorem for the hyperspace of a snake-like continuum. Fund. Math. 50 (1961/62) 237-248. 3. Convergence of inverse systems. Paciﬁc J. Math. 12 (1962) 371-374. 4. Mapping norms and indecomposability. J. London Math. Soc. 39 (1964) 598-602. 5. Isomorphic complexes. Bull. Amer. Math. Soc. 71 (1965) 571-572. ACADEMIC DESCENDANTS AND PUBLICATIONS 247

6. Generalized Frechet varieties. Fund. Math. 58 (1966) 31-43. 7. Isomorphic complexes. II. Bull. Amer. Math. Soc. 72 (1966) 300302. 8. Ouasi dimension type. II. Types in l-dimensional spaces. Pacific J. Math. 25 (1968) 353-370. 9. Local connectedness of inverse limit spaces. Duke Math. J. 28 (1961) 253-259. (with Fort, M. K., Jr.) 10. Minimal representations of the hyperspace of a continuum. Duke Math. J. 32 (1965) 129-137. 11. xxxx-mappings onto polyhedra. Trans. Amer. Math. Soc. 109 (1963) 146-164. (with Mardesic, Sibe) 12. A note on polyhedra embeddable in a plane. Duke Math. J. 33 (1966) 633-638. 13. Ouasi dimension type. I. Types in the real line. Pacific J. Math. 20 (1967) 501-534. 14. E-mappings and generalized manifolds. Mich. Math. J. 14 (1967) 171-182. (with Mardesic, Sibe) 15. On polyhedra embeddable in the 2-sphere. 01 asnik Mat. Ser. III 1 (21) (1966) 167-175. 16. -mappings and generalized manifolds. II. Mich. Math. J. 14 (1967) 423-426. (with Mardesic, Sibe) 17. Isomorphic cone-complexes. Pacific J. Math. 22 (1967) 345-348. (with Thomas, E. S. Jr.)

Kasriel, Robert H. 02 • 1. k-fold irreducible decomposition of a space relative to a mapping. Proc. Amer. Math. Soc. 5 (1954) 440-446. 2. Stability of certain quasi-open mappings. Duke Math. J. 28 (1961) 595-605. 3. A characterization of certain quasi-open mappings. Proc. Amer. Math. Soc. 13 (1962) 778-783. 4. On contractive mappings in uniform spaces. Proc. Amer. Math. Soc. 15 (1964) 288-290. (with Kammerer, William J.) 5. Stability of solutions of some classes of nonlinear operator equations. Proc. Amer. Math. Soc. 17 (1966) 1036-1042. (with Nashed, M. Z.) 248 CHAPTER 9

Cain, George L. 03 • 1. Compact and related mappings. Duke Math. J. 33 (1966) 639- 645. 2. Compactiﬁcation of mappings. Proc. Amer. Math. Soc. 23 (1969) 298-303.

Fuller, Richard V. 03 • 1. Relations among continuous and various non-continuous functions. Paciﬁc J. Math. 25 (1968) 495-509.

Brown, David L. 03 • 1. The conditional level of student’s t test. Ann. Math. Statist. 38 (l967) 1068-1071. 2. Suﬃcient statistics in the case of independent random variables. Ann. Math. Statist. 35 (1964) 1456-1474. 3. On the admissibility of invariant estimators of one or more location parameters. Ann. Math. Statist. 37 (1966) 1087-1136.

Plunkett, Robert L. 02 • 1. On the convergence of matrix iteration processes. Quart. Appl. Math. 7 (1950) 419-421. 2. On the rate of convergence of relaxation methods. Quart. Appl. Math. 7 (1952) 263-266. 3. Representatives of homotopy classes of mappings into spheres. Duke Math. J. 21 (1954) 599-605. 4. A theorem about mappings of a topological group into the circle. Mich. Math. J. 2 (1954) 123-125. 5. Some implications of semi-l-connectedness. Proc. Amer. Math. Soc. 5 (1954) 665-670. 6. A ﬁxed point theorem for continuous multi-values transformations. Proc. Amer. Math. Soc. 7 (1956) 160-163. 7. A topological proof of a theorem of complex analysis. Proc. Nat. Acad. Sci. 42 (1956) 425-426. 8. Concerning two types of convexity for metrics. Arch. Math. 10 (1959) 42-45. ACADEMIC DESCENDANTS AND PUBLICATIONS 249

9. A topological proof of the continuiuty of the derivative of a function complex variable. Bull. Amer. Math. Soc. 65 (1959) 1-4. 10. Openness of the derivative of a complex function. Proc. Amer. Math. Soc. 11 (1960) 671-675. 11. On the contract problem of thih circular rings. Trans. ASME Ser. E. J. Appl. Mech. 32 (1965) 11-20. (with Wu, Chien- Heng.)

Morris, Joseph R. 03 • 1. Queues for a vehicle-actuated traﬃc light. Operations Res. 12 (1964) 882-895. (with Darroch, J. N. and Newell, G. F.)

Williams, Robert F. 02 • 1. Local contractions of compact metric sets which are not local isometries. Proc. Amer. Math. Soc. 5 (1954) 652-654. 2. Local properties of open mappings. Duke Math. J. 22 (1955) 339345. 3. A note on unstable homeomorphisms. Proc. Amer. Math. Soc. 6 (1955) 308-309. 4. Reduction of open mappings. ibid. 7 (1956) 312-318. 5. The effect of maps upon the dimension of subsets of the domain space. ibid. 8 (1957) 580-583. 6. Lebesgue area of maps from Hausdorff spaces. Acta. Math. 102 (1959) 33-46. 7. Local contractions and the size of a compact metric space. Duke Math. J. 26 (1959) 277-289. 8. Lebesgue area zero, dimension and fine-cyclic elements. Riv. Mat. Univ. Parma 10 (1959) 131-143. 9. A useful functor and three famous examples in topology. Trans. Amer. Math. Soc. 106 (1963) 319-329. 10. The construction of certain o-dimension transformation groups. Trans. Amer. Math. Soc. 129 (1967) 140-156. 11. One-dimensional non-wandering sets. Topology 6 (1967) 473- 487. 12. The zeta function of an attractor. Conference on the Topology of Manifolds (Michigan State Univ. E. Lansing, Mich., 1967), pp. 155-161. Prindle, Weber, & Schmidt, Boston, Mass., 1968. 250 CHAPTER 9

13. Compact non-Lie groups. Proc. Conf. on Transformation Groups. (New Orleans, La., 1967), pp. 366-369. Springer, New York, 1968. 14. p-adic groups of transformations. Trans. Amer. Math. Soc. 99 (1961) 488-498. (with Bredon, G. E.; Raymond, Frank) 15. Examples of p-adic transformation groups. Bull. Amer. Math. Soc. 66 (1960) 392-394. (with Raymond, Frank) 16. Examples of p-adic transformation groups. Ann. of Math. (2) 78 (1963) 92-106. 17. Future stability is not generic. Proc. Amer. Math. Soc. 22 (1969) 482-484. (with Shub, Michael)

McDougle, Paul 02 • 1. A theorem on quasi-compact mappings. Proc. Amer. Math. Soc. 9 (1958) 474-477. 2. Mapping and space relations. Proc. Amer. Math. Soc. 10 (1959) 320-323.

Duda, Edwin 02 • 1. Brouwer property spaces. Duke Math. J. 30 (1963) 647-660. 2. A locally compact separable metric space is almost invariant under a closed mapping. Bull. Amer. Math. Soc. 70 (1964) 285-286. 3. Mappings on spheres. Fund. Math. 55 (1964) 195-197. 4. A locally compact metric space is almost invariant under a closed mapping. Proc. Amer. Math. Soc. 16 (1965) 473-475. 5. Reflexive compact mappings. Proc. Amer. Math. Soc. 17 (1966) 688-693. 6. A theorem on one-to-one mappings. Pacific J. Math. 19 (1966) 253-357. 7. Compactness of mappings. Pacific J. Math. 29 (1969) 259-266.

Keesling, J. E. 03 • 1. Mappings and dimensions in general metric spaces. Paciﬁc J. Math. 25 (1968) 277-288. 2. Open and closed mappings and compactiﬁcation. Fund. Math. 20 (1969) 73-81. ACADEMIC DESCENDANTS AND PUBLICATIONS 251

3. Closed mappings which lower dimension. Colloq. Math. 20 (1969) 237-241. 4. Normality and properties related to compactness in hyperspaces. Proc. Amer. Math. Soc. 24 (1970) 760-766. 5. Normality and compactness are equivalent in hyperspaces. Bull. Amer. Math. Soc. 76 (1970) 618-619.

Williams, G. K. 02 • 1. Non-analytic functions of two complex variables. Duke Math. J. 34 (1967) 249-254. 2. On continuity in two variables. Proc. Amer. Math. Soc. 23 (1969) 580-582.

Duke R. A. 02 • 1. Open mappings on graphs and manifolds. Fund. Math. 60 (1967) 149-155. 2. An irreducible graph consisting of a single block. J. Math. Mech. 15 (1966) 129-135. (with Brown, T. A.) 3. The genus, regional number, and Betti number of a graph. Canad. J. Math. 18 (1966) 817-822.

Dickman, R. F. 02 • 1. Unicoherence and related properties. Duke Math. J. 34 (L967) 343-351. 2. Compactness of mappings on products of locally connected generalized continua. Proc. Amer. Math. Soc. 18 (1967) 1093- 1094. 3. Some characterizations of the Freudenthal compactiﬁcation of a semicompact space. Proc. Amer. Math. Soc. 19 (1968) 631-633. 4. Uniﬁed spaces and singular sets for mappings of locally compact spaces. Fund. Math. 62 (1968) 103-123. 5. On openness properties of mappings. Portugal. Math. 26 (1967) 115-123. 6. A theorem on one-to-one mappings onto the plane. Proc. Amer. Math. Soc. 21 (1969) 119-120. 252 CHAPTER 9

7. Semigroups and clusters of indecomposability. Fund. Math. 56 (1964) 21-33. (with Kelley, J. L.; Rubin, L. R. and Swingle, P. M.) 8. Characterization of n-spheres by an excluded middle membrane principle. Mich. Math. J. 11 (1964) 53-59. (with Rubin, L. R. and Swingle, P. M.) 9. Another characterization of n-sphere and related results. Paciﬁc J. Math. 14 (1964) 871-878. (with Rubin, L. R. and Swingle, P. M.) 10. Irreducible continua and generalization of hereditarily unicoherent continua by means of membranes. J. Austral. Math. Soc. 5 (1965) 416-426. (with Rubin, L. R. and Swingle, P. M.)

Garcia-Maynez, A. C. 02 • 1. A factorization theorem for continuous functions. An. Inst. Mat. Univ. Nac. Autonoma Mexico 4 (1964) 33-41. 2. A generalization of the theorem of Feuerbach. Acta Mexicana Ci. Tecn. 3 (1969) 53-57.

McMillan, Evelyn 02 • 1. Asympototic values of functions holomorphic in the unit disc. Mich. Math. J. 12 (1965) 141-154. 2. Principal cluster values of continuous functions. Math Zeit. 91 (1966) 186-197. 3. On radial limits and uniqueness of holomorphic functions. Math. Zeit. 92 (1966) 321-322.

Wallace, Alexander Doniphan 02 • 1. Some invariants under monotone transformation. Bull. Amer. Math. Soc. 45 (1939) 294-295. (with Hall, D. H.) 2. Some characterizations of interior transformations. Amer. J. Math. 51 (1939) 757-763. 3. On non-boundary sets. Bull. Amer. Math. Soc. 45 (1939) 420-422. 4. Concerning relatively non-alternating transformations. Proc. Math. Acad. Sci. 27 (1938) 81-84. ACADEMIC DESCENDANTS AND PUBLICATIONS 253

5. Monotone coverings and monotone transformations. Duke Math. J. 6 (1940) 81-84. 6. O-regular transformations. Amer. J. Math. 52 (1940) 227-284. 7. Quasi monotone transformations. Duke Math. J. 7 (1940) 316- 345. 8. Separation spaces. Ann. of Math. 42 (1941) 687-697. 9. A ﬁxed-point theorem for trees. Bull. Amer. Math. Soc. 47 (1941) 757-760. 10. The acyclic elements of a Peano space. Bull. Amer. Math. Soc. 47 (1941) 778-780. 11. Reducible properties of Peano spaces. Ann. de Acad. Bras. Cien. 14 (1942) 47-49. 12. Separation spaces II. Ann.de Acad. Bras. Cien. 14 (1942) 203- 206. 13. Monotone transformations. Duke Math. J. 9 (1942) 487-506. 14. A substitute for the axiom of choice. Bull. Amer. Math. Soc. 50 (1944) 278. 15. A ﬁxed point theorem. Bull. Amer. Math. Soc. 51 (1945) 613-616. 16. Dimensional types. Bull. Amer. Math. Soc. 51 (1945) 679-681. 17. Generalized arc-sets. Proc. Nat. Acad. Sci. 31 (1945) 414-417. 18. Extension sets. Trans. Amer. Math. Soc. 59 (1946) 1-13. 19. Group invariant continua. Fund. Math. (1949) 119-124. 20. Endelements and the inversion ofcontinua. Duke Math. J. 16 (1949) 141-144. 21. Cyclic invariance under multi-valued maps. Bull. Amer. Math. Soc. 55 (1949) 820-824. 22. Cohomology groups near a set. Ann. de Acad. Bras. Cien. 22 (1950) 217225. 23. A theorem on endpoints. Ann. de Acad. Bras. Cien. 22 (1950) 30-33. 24. Extensional invariance. Trans. Amer. Math. Soc. 70 (1951) 97 102. 25. The map excision theorem. Duke Math. J. 22 (1952) 177-182. 26. A note on mobs I. Ann. de Acad. Bras. Cien. 24 (1952) 229-334. 254 CHAPTER 9

27. Boolean rings and cohomology. Proc. Amer. Math. Soc. 4 (1953) 475. 28. Cohomology, dimension and mobs. Summa Bras. Math. 3 (1953) 53-54. 29. Indecomposable semigroups. Math. J. Okayama University 3 (1953) 1-3. 30. Inverse in Euclidean mobs. Math. J. Okayama University 3 (1953) 335-336. 31. Maximal ideals in compact semigroups. Duke Math. J. 21 (1954) 681-686. 32. A note on mobs II. Ann. de Acad. Bras. Cien. 25 (1953) 335-336. 33. Topological invariance of ideals in mobs. Proc. Amer. Math. Soc. 5 (1954) 866-868. 34. The structure of topological semigroups. Bull. Amer. Math. Soc. 61 (1955) 95-112. (Invited address before annual meeting of AMS, Baltimore, Md., Dec. 28, 1953.) 35. Struct ideals. Proc. Amer. Math. Soc. 4 (1955) 634-638. 36. The position of C-sets in semigroups. Proc. Amer. Math. Soc. 4 (1955) 639-642. 37. One-dimensional clans are groups. Indag. Math.. 17 (1955) 578-580. 38. The Rees-Suschkewitsch structure theorem for compact simple semigroups. Proc. Nat. Acad. Sci. 42 (1956) 430-432. 39. The Gebieststrueue in semigroups. Indag. Math. 18 (1956) 271-274. 40. Ideals in compact connected semigroups. Indag. Math. 18 (1956) 535-539. 41. The center of a compact lattice is totally disconnected. Pacific J. Math. 7 (1957) 1237-1238. 42. Two theorems in topological lattices. Pacific J. Math. 7 (1957) 12391241. 43. The peripheral character of central elements of a lattice. Proc. Amer. Math. Soc. 8 (1957) 596-597. 44. Stability in semigroups. Duke Math. J. 24 (1957) 193-196. 45. Retractions in semigroups. Pacific J. Math. 7 (1957) 1513-1517. ACADEMIC DESCENDANTS AND PUBLICATIONS 255

46. Factoring a lattice. Proc. Amer. Math. Soc. 9 (1958) 250-252. 47. Admissibility of semigroup structures on continua. Trans. Amer. Math. Soc. 88 (1958) 277-287. (with Koch, R. J.) 48. Remarks on aﬃne semigroups. Bull. Amer. Math. Soc. 66 (1960) 110-112. 49. Acyclicity of compact connected semigroups. Fund. Math. 50 (1961) 123-124. 50. A theorem on acyclicity. Bull. Amer. Math. Soc. 67 (1961) 123-124. 51. Relative invertibility in semigroups. Czech. Math. J. 11 (1961) 480-482. 52. A note on convexity. Colloq. Math. 8 (1961) 223-224. 53. . Problems on semigroups. Colloq. Math. 8 (1961) 237-238. 54. A local property of pointwise periodic homomorphism. Colloq. Math. 9 (1962) 64-65. 55. Relative ideals in semigroups (Faucett’s theorem). Colloq. Math. 9 (1962) 55-61. 56. Relative ideals in semigroups. II. Acta. Math. 14 (1963) 137- 148. 57. Notes on inverse semigroups. 58. A note on generalizations of transitive systems of transformations. Colloq. Math. VII (1967) 29-34. (with Aczel, J.) 59. Some theorems of P-intersective sets. Acta. Math. Acad. Sci. Hungar. 17 (1966) 9-14. (with Bednarek, A. R.) 60. Relative ideals and their complements. I. Rev. Math. Pures et Appl. (1966) 13-22.(with Bednarek, A. R.) 61. Semigroups acting on compact totally disconnected spaces. (with Bednarek, A. R.) 62. Semigroups acting on continua. J. Austral. Math. Soc. VII, No. 3 (1967) 327-340.(with Day, J. M.) 63. Problems on periodicity: Functions and semigroups (in memo- riam G. Schweigert). Math. Fyzikalny Casopis, Rocnik 16, Coslo 3 (1966) 209-212. 64. Relations on Topological Spaces. Proc. Symp., Prague, 1961. 65. Problems concerning semigroups. Bull. Amer. Math. Soc. 68 No. 5 (1962) 447-448. 256 CHAPTER 9

66. The least element map. Instytut Math. Polski Akad. Nauk XV, No. 2 (1966) 217-221.(with Franklin, S. P.) 67. A relation-theoretic result with application in topological algebra. Mathematical Systems Theory 1, No. 3 (1966) 217-224. (with Bednarek, A. R.) 68. Finite approximants of compact totally disconnected machines. Mathematical Systems Theory 1, No. 3 (1967) 209-216.(with Bednarek, A. R.) 69. Equivalence on machine state spaces. Mat. Casopis. I 17 No. 1 (1967) 1-7.(with Bednarek, A. R.) 70. Recursions with semigroup state-spaces. Rev. Roum. de Mat. Pures et Appl. XII (1967) 1411-1415. 71. Multiplication induced in the state-space of an act. Mathemat- ical Systems Theory 1, No. 4 (1967) 305-314.(with Day, J. M.) 72. Externally induced operations (submitted to Mathematical Sys- tems Theory, April 26, 1968). 73. Recent results on binary topological algebra. Reprinted from SEMIGROUPS. Academic Press, Inc., 1969, 261-267. 74. Externally induced operations. Math. Systems Theory 3 (1969) No. 3 244-245. 75. Dispersal in compact semimodules, accepted by the J. Austral. Math. Soc. 76. Externally induced operations,Jahr. der Deut. Mat. Ver..

Saalfrank C. W. 03 • 1. Retraction properties for normal Hausdorff spaces. Fund. Math. 36 (1949) 93-10. 2. On the universal covering space and the fundamental group. Proc. Amer. Math. Soc. 4 (1953) 650-653. 3. Neighborhood retraction generalized for compact Hausdorff spaces. Portugal. Math. 20 (1961) 11-16. 4. A generalization of the concept of absolute retract. Proc. Amer. Math. Soc. 12 (1961) 374-378. 5. A characterization of contractible compact Hausdorff spaces. Portugal. Math. 21 (1962) 374-378.(with Jaworowski, J. W.)

Butcher, George H. 03 • ACADEMIC DESCENDANTS AND PUBLICATIONS 257

1. An extention of the sum theorem of dimension theory. Duke Math. J. 18 (1951) 859-874.

Keesee, John W. 03 • 1. Finitely-valued cohomology groups. Proc. Amer. Math. Soc. 1 (1950) 418-422. 2. On the homotopy axiom. Ann. of Math. (2) 54 (1951) 247-249. 3. Sets which separate spheres. Proc. Amer. Math. Soc. 5 (1954) 193-200.

BraSher, Rus 04 • 1. A separation theorem for manifolds. Proc. Amer. Math. Soc. 23 (1969) 242-245. 2. The homology sequence of the double covering: Betti numbers and duality. Proc. Amer. Math. Soc. 23 (1969) 714-Z17.

Strother, Waymon L. 03 • 1. On an open question concerning fixed points. Proc. Amer. Math. Soc. 4 (1953) 988-993. 2. Multi-homotopy. Duke Math. J. 22 (1955) 281-285. 3. Fixed poins, fixed sets, and M-retracts. ibid. 22 (1955) 551-556. 4. Continuous multivalued functions. Bol. Soc. Math. Sao Paulo. 10 (1955) 87-120 (1958). 5. A space of subsets having the fixed point property. Proc. Amer. Math. Soc. 7 (1956) 707-708.(with Capel , C. E.) 6. A theorem of Hamilton: Counter example. Duke Math. J. 24 (1967) 57. 7. Multi-valued functions and partial order. Portugal. Math. 17 (1958) 41-47. (with Capel, C. E.) 8. Retracts from neighborhood retracts. ibid. 25 (1957) 11-14. (with Ward, L. E., Jr.)

Cohen, Haskell 03 • 1. A cohomological deﬁnition of dimension for locally compact Haus- dorﬀ spaces. Duke Math. J. 21 (1954) 209-224. 258 CHAPTER 9

2. Fixed points in products of ordered spaces. Proc. Amer. Math. Soc. 7 (1956) 703-706. 3. A correction. Duke Math. J. 25 (1958) 601. 4. A clan with zero without the fixed point property. Proc. Amer. Math. Soc. 11 (1960) 937-939. (with Collins, H. S.) 5. Affine semigroups. Trans. Amer. Math. Soc. 93 (1969) 97-113. (with Collins, H. S.) 6. Acyclic semigroups and multiplications on two-manifolds. Trans. Amer. Math. Soc. 118 (1965) 420-427.(with Koch, R. J.) 7. On fixed points of commuting functions. Proc. Amer. Math. Soc. 15 (1964) 293-296. 8. Non-Euclidean points in sphere-like spaces. Math Zeit. 107 (1968) 129-132.(with Koch, R. J.) 9. Continuous homomorphic images of real clans with zero. Proc. Amer. Math. Soc. 10 (1959) 106-109. (with Krule, I. S.) 10. On tevates of continuous functions on a unit ball. Proc. Amer. Math. Soc. 18 (1967) 408-411.(with Machigian, Jack.) 11. Clans with zero on an interval. Trans. Amer. Math. Soc. 88 (1958) 523-535.(with Wade, L. I.)

Gordon, William L. 03 • 1. On the coeﬃcient group in cohomology. Duke Math. J. 21 (1954) 139-153. 2. Locally-ﬁnitely-valued cohomology groups. Proc. Amer. Math. Soc. 6 (1955) 656-662. 3. A theorem on uniform Cauchy points. Amer. J. Math. 74 (1952) 764-768. (with McArthur, C. W.)

Ward, Lee E., Jr. 03 • 1. Geodesics and plane arcs on an oblate spheriod. Amer. Math. Monthly 50 (1943) 423-429. 2. Binary relations in topological spaces. Ann. de Acad. Brasil. Cien. 26 (1954) 357-373. 3. Partially ordered topological spaces. Proc. Amer. Math. Soc. 5 (1954) 144-161. ACADEMIC DESCENDANTS AND PUBLICATIONS 259

4. A note on dendrites and trees. Proc. Amer. Math. Soc. 5 (1954) 992-994. 5. Continua invariant under monotone transformations. J. London Math. Soc. 31 (1956) 114-119. 6. Completeness in semi-lattices. Canadian J. Math. 9 (1957) 578582. 7. On the optimal allocation of limited resources. Operations Res. 5 (1967) 815-819. 8. Mobs, trees, and ﬁxed points. Proc. Amer. Math. Soc. 8 (1957) 798-804. 9. On dendritic sets. Duke Math. J. 25 (1958) 505-513. 10. Ring homomorphisms which are also lattice homomorphisms. Amer. J. Math. 61 (1939) 783-787. 11. Note on the general rational solution of the equation ax2 - by2 = z3. Amer. J. Math. 61 (1939) 788-790. 12. The arithmetical properties of modular lattices. Revista Ci. Lima 41 (1939) 593-603. 13. Residuated distributive lattices. Duke Math. J. 6 (1940) 641- 651. 14. The closure operators of a lattice. Ann. of Math. (2) 43 (1942) 191-196. 15. Euler’s three biquadrate problem. Proc. Nat. Acad. Sci. 31 (1945) 125-127. 16. Memoir on elliptic divisibility sequences. Amer. J. Math. 70 (1948) 31-74. 17. Euler’s problem on sums of three fourth powers. Duke Math. J. 15 (1948) 827-837. 18. The law of repetition of primes in an elliptic divisibility sequence. Duke Math. J. 15 (1948) 941-946. 19. Arithmetical properties of the elliptic polynomials arising from the real multiplication of the Jacobi functions. Amer. J. Math. 72 (1950) 284-302. 20. Arithmetical properties of polynomials associated with the lem- niscate elliptic functions. Proc. Nat. Acad. Sci. 39 (1950) 359-362. 21. A class of soluble Diophantine equations. Proc. Nat. Acad. Sci. 37 (1951) 113-114. 260 CHAPTER 9

22. Cyclotomy and the converse of Fermat’s theorem. Amer. Math. Monthly 61 (1954) 564. 23. The maximal prime divisors of linear recurrences. Canadian J. Math. 6 (1954) 455-462. 24. Prime divisors of second order recurring sequences. Duke Math. J. 21 (1954) 607-614. 25. On the number of vanishing terms in an integral cubic recur- rence. Amer. Math. Monthly 62 (1955) 155-160. 26. The intrinsic divisors of Lehmer numbers. Ann. of Math. (2) 62 (1955) 230-236. 27. The mappings of the positive integers into themselves which preserve division. Pacific J. Math. 5 (1955) 1013-1023. 28. The laws of apparition and repetition of primes in a cubic recur- rence. Trans. Amer. Math. Soc. 79 (1955) 72-90. 29. A fixed point theorem. Amer. Math. Monthly 65 (1958) 271-272. 30. A fixed point theorem for multi-valued functions. Pacific J. Math. 8 (1958) 921-927. 31. A fixed point theorem for chained spaces. Pacific J. Math. 9 (1959) 1273-1278. 32. On local trees. Proc. Amer. Math. Soc. 11 (1960) 940-944. 33. Characterization of the fixed point property for a class of set- valued mappings. Fund. Math. 50 (1961/62) 159-164. 34. Fixed point theorems for pseudo monotone mappings. Proc. Amer. Math. Soc. 13 (1962) 13-16. 35. Concerning Koch’s theorem on the existence of arcs. Pacific J. Math. 15 (1965) 347-355. 36. A weak Tychonoff theorem and the axiom of choice. Proc. Amer. Math. Soc. 13 (1962) 757-758. 37. On a conjecture of R. J. Koch. Pacific J. Math. 15 (1965) 14291433. A general fixed point theorem. Colloq. Math. 15 (1966) 243-251. 38. On the non-cutpoint existence theorem. Canad. Math. Bull. 11 (1968) 213-216. 39. One dimensional topological semilattices. Illinois J. Math. 5 (1961) 182-186.(with Anderson, L. W.) 40. A structure theorem for topological lattices. Proc. Glasgow Math. AsSoc. 5 (1961) 1-3. ACADEMIC DESCENDANTS AND PUBLICATIONS 261

41. The lattice theory of ova. Ann. of Math. 40 (1939) 600-608. (with Dilworth, R. P.) 42. Retracts from neighborhood retracts. Duke Math. J. 22 (1955) 551-556.(with Strother, W. L.) 43. On three problems of Franklin and Wallace concerning partially ordered spaces. Colloq. Math. 20 (1969) 229-236.(with Tym- chatyn, E. D.)

Smithson, R. E. 04 • 1. Changes of topology and fixed points for multi-valued functions. Proc. Amer. Math. Soc. 16 (1965) 448-454. 2. Some general properties of multi-valued functions. Pacific J. Math. 15 (1965) 681-703. 3. A fixed point theorem for connected multi-valued functions. Amer. Math. Monthly 73 (1966) 351-355. 4. On criteria for continuiy. Nieuw Arch. Wisk. (3) 14 (1966) 89-92. 5. A note on acyclic continua. Colloq. Math. 19 (1968) 67-71. 6. A characterization of lower semi-continuity. Amer. Math. Monthly 75 (1968) 505. 7. A note on the continuity of multifunctions. J. Natur. Sci. and Math. 7 (1967) 197-202. 8. A note on δ-continuity and proximate fixed points for multi- valued functions. Proc. Amer. Math. Soc. 23 (1969) 256-260. 9. Fixed points and proximate fixed points. Fund. Math. 63 (1968) 321-326.(with Muenzenberger, T. R.)

Harris J. K. 04 • 1. A variation on a problem in number theory of H. Steinhaus. ElemO Math. 21 (1966) 60.(with Crittenden, Richard B.)

Lee, Yu-Lee 04 • 1. Some characterizations of semi-locally connected spaces. Proc. Amer. Math. Soc. 16 (1965) 1318-1320. 2. Topologies with the same class of homeomorphisms. Paciﬁc J. Math. 20 (1967) 77-83. 262 CHAPTER 9

3. A characterization of completeness. Kyungpook Math. J. 7 (1967) 5-6. 4. Continua with the same class of homeomorphism. Kyungpook Math. J. 7 (1967) 1-4. 5. A characterization of Baer lower radical property. Kyungpook Math. J. 7 (1967) 45-46. 6. Characterizing the topology by the class of homeomorphisms. Duke Math. J. 35 (1968) 625-629. 7. Homeomorphisms on manifolds. Kyungpook Math. J. 7 (1967) 31-36. 8. On the construction of upper radical properties. Proc. Amer. Math. Soc. 19 (1968) 1165-1166. 9. On a class of finer topologies with the same class of homeomorphisms. Proc. Amer. Math. Soc. 21 (1969) 129-133. 10. On the construction of lower radical properties. Pacific J. Math. 28 (1969) 393-395. 11. A radical coinciding with the lower radical in associative and alternative rings. Pacific J. Math. 30 (1969) 459-462.(with Leav- itt, W. G.) 12. Topological spaces with large uniformities. Kyungpook Math. J. 7 (1967) 7-8.(with Mozzochi, C. J.) 13. Radical properties and partitions of rings. Kyungpook Math. J. 7 1967) 37-39. (with Propes, R. E. 04)

Propes, Richard 05 • 1. Radical properties and partitions of rings. Kyungpook Math. J. 7 (1967) 37-39.(with Lee, Yu-Lee)

Dimitroﬀ, G. E. 04 • 1. Two characterizations of compact local trees. Trans. Amer. Math. Soc. 127 (1967) 204-220.

Ferguson, E. N. 04 • 1. Commutative rims in clans with zero. Proc. Amer. Math. Soc. 23 (1969) 304-305

Tymchatyn, E. D. 04 • ACADEMIC DESCENDANTS AND PUBLICATIONS 263

1. A one-parameter subsemigroup which meets many regular D- classes. Canadian J. Math. 21 (1969) 735-739. 2. The 2-cell as a partially ordered space. Paciﬁc J. Math. 30 (1969) 825-836. 3. On three problems of Franklin and Wallace concerning partialy ordered sets. Colloq. Math. 20 (1969) 229-236.(with Ward, L. E., Jr.)

Mohler, L. K. 04 • 1. A note on hereditarily locally connected continua. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 17 (1969) 699-701. Faucett, William M. Q3 2. Compact semigroups irreducibly connected between two idempotents. Proc. Amer. Math. Soc. 6 (1955) 741-747. 3. Topological semigroups and continua with cut points. ibid. 6 (1955) 748-756. 4. Complements of maximal ideals in compact semigroups. Duke Math. J. 22 (1955) 655-661.(with Koch, R. J.)

Anderson, Lee W. 03 • 1. Topological lattices and n-cells. Duke Math. J. 25 (1958) 205- 208. 2. On the breadth and co-dimension of a topological lattice. Paciﬁc J. Math. 9 (1959) 327-333. 3. One dimensional topological lattices. Proc. Amer. Math. Soc. lO (1959) 715-720. 4. On the distributivity and simple connectivity of plane topological lattices. Trans. Amer. Math. Soc. 91 (1959) 102-112. 5. Locally compact topological lattices. Proc. Sympos. Pure Math., Vol. II, pp. 195-197. American Mathematical Society, Provi- dence, R. I., 1961. 6. The existence of continuous lattice homomorphisms. J. London Math. Soc. 37 (1962) 60-62. 7. The κ-equivalence in compact semigroups. Bull. Soc. Math. Belg. 14 (1962) 274-296.(with Hunter, R. P.) 8. Homomorphism and dimension. Math. Ann. 147 (1962) 248- 268. 264 CHAPTER 9

9. Small continua at certain orbits. Arch. Math. 14 (1963) 350-353. 10. The κ-equivalence in compact semigroups. II. J. Austral. Math. Soc. 3 (1963) 288-293. 11. Sur les espaces ﬁbres associes a une D-class d’un demigroupe compact. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 12 (1964) 249-251.(with Hunter, R. P.) 12. Sur les demi-groupes compacts et connexes. Fund. Math. 56 (1964) 183-187. 13. Une version bilatere du groupe de Schutzenberger. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 13 (1965) 527-531. 14. Certain homomorphisms of a compact semigroup onto a thread. J. Austral. Math. Soc. 7 (1967) 311-322.(with Hunter, R. P.) 15. Groups, Homomorphisms, and the Green relations. Proc. Inter- nat. Conf. Theory of Groups (Canberra, 1965) pp. 1-5. Gordon and Breach, New York, 1967. 16. On residual properties of certain semigroups. Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967) pp. 15-19. Deutsch. Verlag Wissensch., Berlin, 1969. 17. On the compactiﬁcation of certain semigroups. Contributions to Extension Theory of Topological Structures. (Proc. Sympos., Berlin, 1967) pp. 21-27. Deutsch. Verlag Wissensch., Berling, 1969. 18. Some results on stability in semigroups. Trans. Amer. Math. Soc. 117 (1965) 521-529.(with Hunter, R. P.; Koch, R. J.) 19. One dimensional topological semilattices. Illinois J. Math. 5 (1961) 182-186.(with Ward, L. E. Jr.) 20. A structure theorem for topological lattices. Proc. Glasgow Math. Assoc. 5 (1961) 1-3.

Stralka, Albert 04 • 1. The Green equivalences and dimension in-compact semigroups. Math. Zeit. 109 (1969) 169-176.

Lin, Y. F. 03 • 1. A note on the Wallace theorem. Portugal. Math. 19 (1960) 199-201. ACADEMIC DESCENDANTS AND PUBLICATIONS 265

2. Characterization of totally non-commutative semigroups. Hung- Ching Chow 60th anniversary Vol., pp. 108-109. Inst. of Math., Acad. Sinica. Taipei, 1962. 3. Generalized character semigroups: The Schwarz decomposition. Pacific J. Math. 15 (1965) 1307-1312. 4. Not necessarily abelian convolution semigroups of probability measure. Math Zeit. 91 (1966) 300-307. 5. A convolution semigroup of modular functions. J. Austral. Math. Soc. 6 (1966) 251-255. 6. Tychonoff’s theorem for the space of multifunctions. Amer. Math. Monthly 74 (1967) 399-400. 7. The structure of not necessarily perfect automata and their au- teomorphism group. I. Hung-Ching Chow 65th Anniversary Vol- ume, pp. 62-71. Math. Res. Center Nat. Taiwan Univ., Taipei, 1967. 8. Relations on topological spaces: Urysohn’s lemma. J. Austral. Math. Soc. 8 (1968) 37-42. 9. A problem of Bosak concerning the graphs of semigroups. Proc. Amer. Math. Soc. 21 (1969) 343-346. 10. A new characterization of Hausdorff n-spaces. Pro. Japan Acad. 44 (1968) 1031-1032.(with Soniat, Leonard.)

Borrego, J. T. 03 • 1. Homomorphic retractions in semigroups. Proc. Amer. Math. Soc. 18 (1967) 716-719. 2. Adjunction semigroups. Bull. Austral. Math. Soc. 1 (1969) 47-58. 3. Point-transitive actions by a standard metric thread. Proc. Amer. Math. Soc. 23 (1969) 261-265. 4. A class of multi-valued functions. Kyungpook Math. J. 9 (1969) 77-78.

Choe Tae-Ho 03 • 1. Notes on the lattice-ordered groups. Kyungpook Math. J. 1 (1958) 37-42. 2. An isolated point in a partly ordered set with interval topology. Kyungpook Math. J. 1 (1958) 57-59. 266 CHAPTER 9

3. The internal topology of a lattice ordered group. Kyungpook Math. J. 1 (1958) 69-74. 4. The topologies of partially ordered set with finite width. Kyung- pook Math. J. 2 (1959) 17-22. 5. On a quasi-ordered group. Kyungpook Math. J. 2 (1959) 47-52. 6. Erratum: Notes on lattice-ordered groups. Kyungpook Math. J. 2 (1959) 73. 7. Notes on the lattice of subgroups of a group. Kyungpook Math. J. 3 (1960) 1-5. 8. Lattice ordered commutative groups of the second kind. Kyung- pook Math. J. 3(1960) 43-48. 9. On the continuity of group operations of an l-group with CP- ideal topology. Kyungpook Math. J. 4 (1961) 13-16. 10. On order-convergence of partially ordered groups. Kyungpook Math. J. 4 (1961) 33-37. 11. On a continuous mapping between partially ordered sets with some topology. Kyungpook Math. J. 5 (1962) 9-13. 12. Notes on the lattice of congruence relations on a lattice. Kyung- pook Math. J. 5 (1962/63) 23-33. 13. An isolated point in a complete lattice with order topology. Kyungpook Math. J. 6 (1964) 1-2. 14. Complement of a congruence relation in a modular lattice. Kyung- pook Math. J. 6 (1964) 3-5. 15. Intrinsic topologies in a topological lattice. Pacific J. Math. 28 (1969) 49-52. 16. On compact topological lattices of finite dimension. Trans. Amer. Math. Soc. 140 (1969) 223-237. 17. The breadth and dimension of a topological lattice. Proc. Amer. Math. Soc. 23 (1969) 82-84. 18. Notes on locally compact connected topological lattices.Canad. J. Math. 21 (1969) 1533-1536.

Sigmon, Kermit N. 03 • 1. A note on means in Peano continua. Aequationes Math. 1 (1968) No. 1-2, 85-86. 2. On the existence of a mean on certain continua. Fund. Math. 63 (1968) 311-319. ACADEMIC DESCENDANTS AND PUBLICATIONS 267

3. Medial topological groupoids. Aequationes Math. 1 (1968) 217234.

Shershin, Anthonoy 03 • 1. A characterization of the closed subgroups of the Schutzenberger group. Acta. Math. Acad. Sci. Hungar. 20 (1969) 179-183. 2. Algebraic results concerning Green’s κ-slices. Colloq. Math. 20 (1969) 221-225. 3. Direct summands of Abelian monoids. Math. Notae 20 (1965) 109-116.(with Moore, John T.)

Chae, Y. 03 • 1. Symmetric spaces which are mapped conformally on each other. Kyungpook Math. J. 2 (1959) 65-72. 2. Gaussian spherical representation of a hypersurface of an Eu- clidean space. I, II. Kyungpook Math. J. 3 (1960) 7-12, 49-54. 3. Conformal collineations in recurrent spaces. Kyungpook Math. J. (1961) 17-21. 4. Projective motives in non-Riemannian K -spaces. I. Kyungpook Math. J. 5 (1962) 15-20.

Koch, R. J. 03 • 1. Remarks on primitive idempotents in compact semigroups with zero. Proc. Amer. Math. Soc. 5 (1954) 16-447. 2. Note on weak cutpoints in clans. Duke Math. J. 24 (1957) 611- 615. 3. On monothetic semigroups. Proc. Amer. Math. Soc. 8 (1957) 397401. 4. Arcs in partially ordered spaces. Paciﬁc J. Math. 9 (1959) 723- 728. 5. Ordered semigroups in partially ordered spaces. Paciﬁc J. Math. 10 (1960) 1333-1336. 6. Threads in compact semigroups. Math Zeit. 86 (1964) 312-316. 7. Connected chains in quasi-ordered spaces. Fund. Math. 56 (1964/65) 245-249. 8. Some open questions in topological semigroups. And. Acad. Brasil. Cien. 41 (1969) 19-20. 268 CHAPTER 9

9. Some results on stability in semigroups. Trans. Amer. Math. Soc. 117 (L965) 521-529.(with Anderson, L. W.; Hunter, R. P.) 10. Acyclic semigroups and multiplications on two-man. Trans. Amer. Math. Soc. 118 (1965) 420-427.(with Cohen, Haskell) 11. Non-Euclidean points in sphere-like spaces. Math Zeit. 107 (1968) 127-132. 12. Regular D-classes in measure semigroups. Trans. Amer. Math. Soc. 105 (1962) 21-31.(with Collins, H. S.) 13. Weak cutpoint ordering on hereditarily unicoherent continua. Proc. Amer. Math. Soc. 11 (1960) 679-681.(with Krule, I. S.) 14. Semigroups on trees. Fund. Math. 59 (1961/62) 341-346.(with McAuley, L. F.) 15. Semi-groups on continua ruled by arcs. Fund. Math. 56 (1964) 1-8. 16. Maximal ideals in compact semigroups. Duke Math. J. 21 (1954) 681-685.(with Wallace, A. D.) 17. Stability in semigroups. Duke Math. J. 24 (1957) 193-195. 18. Admissibility of semigroup structures on continua. Trans. Amer. Math. Soc. 88 (1958) 277-287. 19. Notes on inverse semigroups. Rev. Roumaine Math. Pures Appl. 9 (1964) 19-24.(with Wallace, A. D.)

Rothman, Neal J. 04 • 1. Imbedding of topological semigroups. Math. Ann. 139 (1960) 197-203. 2. Linearly quasi-ordered compact semigroups. Proc. Amer. Math. Soc. 13 (1962) 352-357. 3. Algebraically irreducible semigroups. Duke Math. J. 39 (1963) 511-517. 4. On the uniqueness of character semigroups. Math. Ann. 151 (1963) 346-354. 5. A note on compact rings. Math Zeit. 91 (1966) 179-184. 6. An L1 algebra for certain locally compact topological semigroups. Paciﬁc J. Math. 23 (1967) 143-151. 7. Separating points by semicharacters in topological semigroups. Proc. Amer. Math. Soc. 21 (1969) 235-239.(with Baker, John W.) ACADEMIC DESCENDANTS AND PUBLICATIONS 269

8. An Ll algebra for algebraically irreducible semigroups. Studia Math. 33 (1969) 257-272.(with Bergman, John G.) 9. An Ll algebra for linearly quasi-ordered compact semigroups. Paciﬁc J. Math. 26 (1968) 579-588. 10. Contractibility of certain semigroups. Bull. Amer. Math. Soc. 70 (1964) 756-757.(with Gottlieb, D. H.) 11. Characters and cross sections for certain semigroups. Duke Math. J. 29 (1962) 347-366.(with Hunter, R. P.)

Bergman, John G. 05 • 1. An L1 algebra for algebraically irreducible semigroups. Studia Math. 33 (1969) 257-272.(with Rothman, N. J.) 2. Arcs in partially ordered spaces. Paciﬁc J. Math. 9 (1959) 723- 728. 3. Ordered semigroups in partially ordered spaces. Paciﬁc J. Math. 10 (1960) 1333-1336. 4. Threads in compact semigroups. Math Zeit. 86 (1964) 312-316. 5. Connected chains in quasi-ordered spaces. Fund. Math. 56 (1964/65) 245-249. 6. Some open questions in topological semigroups. Ann. de Acad. Brasil. Cien. 41 (1969) 19-20. 7. Some results on stability in semigroups. Trans. Amer. Math. Soc. 117 (L965) 521-529.(with Anderson, L. W.; Hunter, R. P.) 8. Acyclic semigroups and multiplications on two-man. Trans. Amer. Math. Soc. 118 (1965) 420-427.(with Cohen, Haskell) 9. Non-Euclidean points in sphere-like spaces. Math Zeit. 107 (1968) 127-132. 10. Regular D-classes in measure semigroups. Trans. Amer. Math. Soc. 105 (1962) 21-31.(with Collins, H. S.) 11. Weak cutpoint ordering on hereditarily unicoherent continua. Proc. Amer. Math. Soc. 11 (1960) 679-681.(with Krule, I. S.) 12. Semigroups on trees. Fund. Math. 59 (1961/62) 341-346.(with McAuley, L. F.) 13. Semi-groups on continua ruled by arcs. Fund. Math. 56 (1964) 1-8. 14. Maximal ideals in compact semigroups. Duke Math. J. 21 (1954) 681-685.(with Wallace, A. D.) 270 CHAPTER 9

15. Stability in semigroups. Duke Math. J. 24 (1957) 193-195. 16. Admissibility of semigroup structures on continua. Trans. Amer. Math. Soc. 88 (1958) 277-287. 17. Notes on inverse semigroups. Rev. Roumaine Math. Pures Appl. 9 (1964) 19-24.(with Wallace, A. D.)

Hunter, R. P. 04 • 1. Type of (n, k) adherence and indecomposability Portugal. Math. 15 (1956) 115-122. 2. On the semigroup structure of continua. Trans. Amer. Math. Soc. 93 (1959) 356-368. 3. Certain upper semicontinuous decompositions of a semigroup. Duke Math. J. 27 (1960) 283-289. 4. Note on arcs in semigroups. Fund. Math. 49 (1960/61) 233-245. 5. Certain compact connected semigoups irreducible over a ﬁnite set. (Spanish) Bol. Soc. Math. Mexicana (2) 6 (1961) 52-59. 6. Certain homomorphisms of compact connected semigroups. Duke Math. J. 28 (1961 83-87. 7. On a conjecture of Koch. Proc. Amer. Math. Soc. 12 (1961) 138139. 8. Sur la position des c-ensembles dans les semi-groupes. Bull. Soc. Math. Belg. 14 (1962) 190-195. 9. On one-dimensional semigroups. Math. Ann. 146 (1962) 383- 396. 10. A remark on a decomposition space of Bing. Proc. Amer. Math. Soc. 13 (1962) 777. 11. On the structure of homogroups with applications to the theory of compact connected semigroups. Fund. Math. 52 (1963) 69- 102. 12. Some results on wreath products of semigroups. Bull. Soc. Math. Belg. 18 (1966) 3-16. 13. The κ-equivalence in compact semigroups. Bull. Soc. Math. Belg. 14 (1962) 274-296.(with Anderson, Lee W.) 14. Homomorphisms and dimension. Math. Ann. 147 (1962) 248- 268. 15. Small continua at certain orbits. Arch. Math. 14 (1963) 350-353. ACADEMIC DESCENDANTS AND PUBLICATIONS 271

16. The κ-equivalence in a compact semigroup. II. J. Austral. Math. Soc. 3 (1963) 288-293. 17. Sur les espaces ﬁbres associes a une D-classes d’un semigroupe compact. Bull. Aca. Polon. Sci. Ser. Sci. Math. Astron. Phys. 12 (1964) 249-251. 18. Sur les demi-grous compacts et connexes. Fund. Math. 46 (1964) 183-187. 19. Une versicn bilatere du groupe de Schutzenberger. Bull. Acad. Polon. Sci. Ser. Sci. Math. Aston. Phys. 13 (1965) 527-531. 20. Certain homomorphisms of a compact semigroup onto a thread. J. Austral. Math. Soc. 7 (1967) 311-322.(with Anderson, L. W.) 21. Groups, homomorphisms, and the Green relations. Proc. Intern. Conf. Theory of Groups (Canberra, 1965) pp. 1-5. Gordon and Breach, New York, 1967. 22. On residual properties of certain semigroups. Contributions to Extension Theory of Topological Structures (Proc. Sympos., Br- lin, 1967) pp. 15-19. Deutsch. Verlag Wissensch., Berlin, 1969. 23. On the compactiﬁcation of certain semigroups. Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967) pp. 21-27. Deutsch. Verlag Wissensch., Berlin, 1969. 24. Some results on stability in semigroups. Trans. Amer. Math. Soc. 117 (1965) 521-529. (with Anderson, Lee W.; Koch, R. J.) 25. Characters and cross sections for certain semigroups. Duke Math. J. 29 (1962) 347-366.(with Rothman, N. J.) 26. Indecomposable trajectories. Tohoku Math. J.(2) 10 (1958) 3- 10. (with Swingle, Paul M.)

Selden, John 05 • 1. A note on compact semirings. Proc. Amer. Math. Soc. 15 (1964) 882-886. 2. Clans on group-supporting spaces. Proc. Amer. Math. Soc. 18 (1967) 540-545.(with Madison, B.) 3. On the closure of the bicyclic semigroup. Trans. Amer. Math. Soc. 144 (1969) 115-126.(with Eberhart, Carl.)

Madison, Bernard 06 272 CHAPTER 9

1. Semigroups on coset spaces. Duke Math. J. 36 (1969) 61-63. 2. Clans on group-supporting spaces. Proc. Amer. Math. Soc. 18 (1967) 540-545.(with Selden, J.)

Stepp, James W. 06 • 1. A note on maximal locally compact semigroups. Proc. Amer. Math. Soc. 29 (1969) 251-253. 2. D-semigroups. Proc. Amer. Math. Soc. 22 (1969) 402-406.

Stadtlander, David 05 • 1. Semigroups, continua and the set functions T . Duke Math. J. 29 (1962) 265-280.(with Davis, H. S., Swingle, P. M.) 2. Properties of the set function T . Portugal. Math. 21 (1962) 113-133. 3. Further properties of the set function T. J. Natur. Sci. and Math. 7 (1967) 91-94. 4. Thread actions. Duke Math. J. 35 (1968) 483-490. 5. Semigroup actions and dimension. Aequationes Math. 3 (1969) 1-14.

Brown, Dennison R. 04 • 1. On clans of non-negative matrices. Proc. Amer. Math. Soc. 15 (1964) 671-674. 2. Topological semilattices on the two-cell. Paciﬁc J. Math. 15 (1965) 35-46. 3. Matrix representations of compact simple semigroups. Duke Math. J. 33 (1966) 69-73. 4. Representation theorems for uniquely divisible semigroups. Duke Math. J. 35 (1968) 341-352.(with Friedberg, Michael) 5. A new notion of semicharacters. Trans. Amer. Math. Soc. 141 (1969) 87-401. 6. A chafacterization of niquely dlvisible commutative groups. Pa- ciﬁc J. Math. 18 (19xxx) 57-60.(with La Tarrej J. G.)

Hildebrandt. J. 05 • ACADEMIC DESCENDANTS AND PUBLICATIONS 273

1. On compact unithetic semigroups. Paciﬁc J. Math. 21 (1967) 265273. 2. On uniquely divisible semigroups on the two-cell. Paciﬁc J. Math. 23 1967) 91-95. 3. On compact divisible Abelian emigroups. Proc. Amer. Math. Soc. 19 tl968) 405-410. 4. The universal compact subunithetic semigroup. Proc. Amer. Math. Soc. 23 (1969) 220-224.

Lawson, Jimmie 05 • 1. A generalized version of the Victoria-Begle theorem. Fund. Math. 65 1969) 65-72. 2. Topological semilattices with small semilattices. J. London Math. Soc. (2) 1 (1969) 719-724.

Friedberg, Michael 04 • 1. A note on Haar-like measure for group-externai semigroups. Proc. Amer. Math. Soc. 17 (1966) 1079-1082. 2. On representations of certain semigroups. Paciﬁc J. Math. 19 (1966) 269-274. 3. Representation theorems for uniquely divisible semigroups. Duke Math. J. 3 (1968) 341-352.(with Brown, Dennison R.) 4. A new notion of semicharacters. Trans. Amer. Math. Soc. 141 (1969) 387-401.(with Brown, Dennison R.) 5. Semigroups on chainable and circle-like continua. Math Zeit. 106 (19683 158-161.(with Mahavier, W. S.)

Clark C. E. 04 • 1. Sampling eﬃciency in Monte Carlo analyses, Ann. Inst. Statist. Math. 15 (1963) 197-206. 2. Locally algebraic independent collections of subsemigroups of semigroups. Duke Math. J. 35 (1968) 843-851. 3. Monotone homomorphisms of compact semigroups. J. Austral. Math. Soc. 9 (1969) 167-175. 4. A characterization of compac connected planar lattices. Paciﬁc J. Math. 24 (1968) 233-240.(with Eberhart, Carl) 274 CHAPTER 9

Carruth, J. H. 04 • 1. A note on partially ordered compata. Paciﬁc J. Math. 24 (1968) 229-231. 2. Lifting trees under light open maps. Fund. Math. 66 (1969/70) 215-217.

Eberhart, Carl 04 • 1. A unique factorization theorem for countable products of circles. Fund. Math. 61 (1967-68) 305-308. 2. Tychonoﬀ cubes are coset spaces. Proc. Amer. Math. Soc. 19 (1968) 185-188. 3. A note on smooth fans. Colloq. Math. 20 (1969) 89-90. 4. Metrizability of trees. Fund. Math. 65 (1969) 43-50. 5. A characterization of compact connected planar lattices. Paciﬁc J. Math. 24 (1968) 233-240.(with Clark, Charles E.) 6. On the closure of the bicyclic subgroup. Trans. Amer. Math. Soc. 144 (1969) 115-126.(with Selden, John)

L’Heureaux J H. 04 • 1. A note on principal normal inverse sub-semigroups. J. Natur. Sci and Math. 6 (1966) 231-233.

Whyburn, G. T. 01 • 1. Dynamic Topology. Amer. Math. Monthly 77 (1970) 556-570. 2. Inward motions in connected spaces. Proc. Nat. Acad. Sci. 63 (1969) 271-274. 3. Functional movements in dendric structures. General Topology and its Relations to Modern Analysis and Algebra, III ( Proc. Conf., Kanpur, 1968) 319-327. Academie, Prague, 1971.

Wallace, A. D. 02 • 1. Recursions with semigroup state-spaces. Rev. Roum. Math. Pure Appl. 12 (1967) 1411-1415. 2. Recent results on binary topological algebras. Semigroups ( Proc. Sympos. Wayne State University, Detroit, Michigan, 1968) 261-267. ACADEMIC DESCENDANTS AND PUBLICATIONS 275

DeVun, Esmond E. 04 • 1. Point-transitive actions by the unit interval. Canadian J. Math. 22 (1970) 255-259.(with Borrego, J. T.)

Koch, Robert J. 03 • 1. Compact connected spaces supporting topological semigroups. Semigroup Forum 1 (1970) no. 3, 209-223. 2. A survey of results on threads. Semigroups. ( Proc. Sympos. Wayne State University, Detroit, Michigan, 1968) 101-106. Aca- demic Press, New York, 1969.

Rothman, Neal J. 04 • 1. Remarks on duality and semigroups. Israel J. Math. 8 (1970) 83-89.

Hunter, Robert P. 04 • 1. A remark on compact semigroups having certain decomposition spaces embeddable in the plane. Bull. Austral. Math. Soc. 4 (1971) 137-139.(with Anderson, Lee) 2. Compact semigroups having certain one-dimensional hyperspaces. Amer. J. Math. 92 (1970) 894-896. 3. Sur les homomorphismes des demi-groupes compacts. Semi- naire P. Dubreil, M. L. Dubreil-Jacotin, L. Leisieiur et C. Pisot: 1968/69, Algebrae et Theorie des Nombres, Fasc. 2, Exp. 20, 5 pp. Secretariat mathematique, Paris, 1970. 4. On the continuiuty of certain homomorphisms of compact semigroups. Duke Math. J. 38 (1971) 409-414.(with Anderson, Lee W.) 5. Corrigendum: ‘A remark on compact semigroups having certain decomposition spaces embeddable in the plane.’ Bull. Austral. Math. Soc. (1971) 432.

Madison, Bernard 06 • 1. A note on local homogeneity and stability. Fund. Math. 66 (1969/70) 123-127. 2. Peripherality in semigroups. Semigroup Forum 1 (1970) no. 2, 128-142.(with Lawson, Jimmie D.) 276 CHAPTER 9

3. Peripheral and inner points. Fund. Math. 69 (1970) 253-266. (with Lawson, Jimmie D.)

Stepp, James W. 06 • 1. Locally compact Cliﬀord semigroups. Paciﬁc J. Math. 34 (1970) 163-176. 2. Semilattices which are embeddable in a product of min intervals. Proc. Amer. Math. Soc. 28 (1971) 81-86. 3. Semigroup extensions. J. Reine Angew. Math. 248 (1971) 28- 41.(with Fulp, R. O.) 4. Structure of the semigroup of semigroup extensions. Trans. Amer. Math. Soc. 158 (1971) 63-73.(with Fulp, R. O.)

Stadtlander, David 05 • 1. Actions with topologically restricted state spaces. Duke Math. J. 37 (1970) 199-206. 2. Actions with topologically restricted state spaces. II Duke Math. J. 38 (1971) 7-14

Brown, Dennison R. 04 • 1. A survey of compact divisible commutative semigroups. Semi- grou Forum I (1970) no. 2, 143-161.(with Friedberg, Michael)

Lawson, Jimmie D. 05 • 1. The relation of breadth and co-dimensions in topological semilattices. Duke Math. J. 37 (1970) 207-212. 2. Lattices with no interval homomorphisms. Paciﬁc J. Math. 32 (1970) 459-465. 3. On the existence of one-par,ameter semigroup,s. Semigreup For,um 1 1970? no. 1, 85=90.(with Carruth, J. H.) 4. Semigroup through semilattices. Trans. Amer. Math. Soc. 152 (1970j 597-608.(with Carruth, J. H.) 5. Peripherality in semigroup. Semigroup Forum 1 (19703 no. 2, 12,8-142.(with Madison, B.) 6. Topological semilattices and their underlying spaces. Semigroup Forum 1 (1970) no. 3, 209-223.(with Williams, Wiley) ACADEMIC DESCENDANTS AND PUBLICATIONS 277

7. Peripheral and inner points. Fund. Math. 6 (19703 253-266. (with Madison, B.)

Farley, Reuben 05 • 1. Positive Cliﬀord semigroups on the plane. Trans. Amer. Math. Soc. 151 (1970? 353-369.

Friedberg, Michael 04 • 1. A survey of compact divisible commutative semigroups, Semi- group Forum 1 (1970) no. 2, 143-161.(with Brown, D. R.)

Clark C. E. 04 • 1. On certain types of congruence on compact commutative semigroups. Duke Math. J. 37 (1970) 95 1 2. Representations of certain compact semigroups by HL-semigroups. Trans. Amer. Math. Soc. 149 (1970 327-337.(with Carruth, J. H.) 3. Compact totally H qrdered semigroups. Proc. Amer. Math. Soc. 27 (1971) 199-204.(with Carruth, J. H.)

Carruth J. H. 04 • 1. Representations of certain copact semigroups by HL-semigroups. Trans. Amer. Math. Soc. 149 (l970) 327-337.(with Clark, C.E.) 2. Compact totally H ordered semigroups. Proc. Amer. Math. Soc. 27 (1971) 199-204.(with Clark, C.E.)

Eberhart, Carl A. 04 • 1. On smooth dendroids. Fund. Math. 67 (1970) 297-322.(with Charatonik, J. J.)

Williams, W. . 04 • 1. Topological semilattices and their underlying spaces. Semigroup Forum 1 (1970) no. 3? 209-223.(with Lawson, Jimmie D.)

McCharen, J. D. 04 • 1. On exending hgmeomgrphisms to Frechet manifolds. Proc. Amer. Math. Soc, 25 (1970? 283-289.(with Anderson, R. D.) 278 CHAPTER 9

2. Semigroups on ﬁnitely ﬂoored spaces. Trans. Amer. Math. Soc. 156 (1971) 85-89.

Ward, Lee E. 03 • 1. Concerning Qntinuous selections. Proc. Amer. Math. Soc. 25 (1970) 369-374.(with Nadler, Sam) 2. Compact directed spaces. Trans. Amer. Math. Soc. 152 (1970) 145-157. 3. Set-valued mappings on partially ordered spaces. Set Valued Mappings, Selections and Topological Properties of 2x.(Proc. Conf. SUNY, Buﬀalo, New York, 1969) 88-89. Lecture Notes in Mathematics. Vol. 171. Springer, Berlin, 1970. 4. Arcs in hyperspace which are not compact. Proc. Amer. Math. Soc. 28 (1971) 254-258.

Smithson, Raymond E. 04 • 1. Topologies generated by relations. Bull. Austral. Math. Soc. 1 (1969) 297-306. 2. A counterexample to a lemma on the existence of simple reﬁne- ments. Amer. Math. Monthly 7 (1970) 293-294. 3. Fixed points for contractive multifunctions. Proc. Amer. Math. Soc. 27 (1971) 192-194. 4. Fixed points of order-preserving multifunctions. Proc. Amer. Math. Soc. 28 (1971) 304-310.

Lee, Yu-Lee 04 • 1. On completion of measure spaces. Kyungpook Math. J. 10 (1970) 1-2. 2. On the existence of non-comparable homogeneous topologies with the same class of homeomorphisms. Tohoku Math. J. (2) 22 (1970) 499-501.

Propes, Richard 05 • 1. A characertization of the Behrens radical. Kyungpook Math. J. 10 (1970) 49-52.

Feruson, E. N. 04 • ACADEMIC DESCENDANTS AND PUBLICATIONS 279

1. No semigroup on the two-cell with idempotent boundary. Duke Math. J. 37 (1970) 601-607.

Tymchatyn, E. D. 04 • 1. Antichains and products in partially ordered spaces. Trans. Amer. Math. Soc. 146 (1969) 511-520.

Mohler, Lee K. 04 • 1. A ﬁxed point theorem for continua which are hereditarily divisible by points. Fund. Math. 67 (1970) 345-358. 2. A characterization of smoothness in dendroids. Fund. Math. 67 (1970) 369-376. 3. A characterization of local connectedness for generalized continua. Colloq. Math. 21 (1970) 81-85.

Anderson, Lee W. 03 • 1. A remark on compact semigroups having certain decomposition spaces. Bull. Austral. Math. Soc. 4 (1971) 137-139.(with Hunter, R. P.) 2. Compact semigroups having certain one-dimensional hyperspaces. Amer. J. Math. 92 (1970) 894-896. 3. On the continuity of certain homeomorphisms of compact semigroups. Duke Math. J. 38 (1971) 409-414. 4. Corrigendum: ‘A remark on compact semigroups having certain decomposition spaces embeddable in the plane.’ Bull. Austral. Math. Soc. 4 (1971) 432.

Stralka, Albert 04 • 1. Rees-simple groups. J. London Math. Soc. (2) I (1969) 705-708. 2. The topological structure of D-classes. Bull. Austral. Math. Soc. I (1969) 289-295. 3. Locally convex topological lattices. Trans. Amer. Math. Soc. 151 (1970) 629-640. 4. Homomorphisms on connected topological lattices. Duke Math. J. 38 (1971) 483-490.(with Shirley, E. D.)

Lin, Y. F. 03 • 280 CHAPTER 9

1. The graphs of semirings. J. Algebra 14 (1970) 73-82.(with Ratti, J. S.) 2. Ascili’s theorem for spaces of multifunctions. Paciﬁc J. Math. 34 (1970) 741-747. (with Rose, David A.) 3. Connectivity of the graphs of semirings: Lifting and product. Proc. Amer. Math. Soc. 24 (1970) 411-414.(with Ratti, J. S.)

Borrego, J. T., Jr. 03 • 1. Continuity of the operation in a semilattice. Colloq. Math. 21 49-52. 2. Point-transitive actions by the unit interval. Canadian J. Math. 22 (1970) 255-259.(with DeVun, E E)

Choe, Tae Ho 03 • 1. Remarks on topological lattices. Kyungpook Math. J. 9 (1969) 59-62.

Sigmon, Kermit T. 03 • 1. Acyclicity of compact means. Mech. Math. J. 16 (1969) 111-115. 2. Cancellative medial means are arithmetic. Duke Math. J. 39 (1970) 439-445.

Shershin, Anthony C. 03 • 1. Results concerning the Schutzenberger-Wallace theorem. Bol. Soc. Mat. Mexicana (2) 13 (1968) 21-31.

Kelley, John L. 02 • 1. Euler characteristics. Paciﬁc J. Math. 26 (1968) 317-339. (with Spanier, E. H.) 2. Pre-measures on lattices of sets. Math. Ann. 190 (1970/71) 233-241.(with Srinivasan, T. P.)

Fell, J. M. G. 03 • 1. An extension of Mackey’s method to Banach *-algebraic bundles. Memoirs Amer. Math. Soc. No. 90. American Mathematical Society, Providence, Rhode Island, 1969, 168 pp. ACADEMIC DESCENDANTS AND PUBLICATIONS 281

Prosser, Reese T. 03 • 1. Note on metric dimension. Proc. Amer. Math. Soc. 25 (1970) 763-765. 2. Determinable classes of channels. II Indiana Univ. Math. J. 20 (1970/71) 789-806.

Namioka, Isaac 03 • 1. Tensor products of compact convex sets. Paciﬁc J. Math. 31 (1969) 469-480.(with Phelps, R. R.)

Bear, H. S. 03 • 1. Continuous selections of representing measures. Bull. Amer. Math. Soc. 76 (1970) 366-369. 2. Lectures on Gleason points. Lect. Notes in Math., Vol. 121. Springer-Verlag, Berlin-New York, 1970, iii+47 pp. 3. The part metrics in convex sets. Paciﬁc J. Math. 30 (1969) 15-33.

Singh, V. 03 • 1. An aspect of local property of the absolute summability of the conjugate series of the r-th derived series of Fourier series. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 17 (1969) 739-744. 2. Upper bounds on the elastic diﬀerential cross sections. Ann. Physics 57 (1970) 461-480.(with Roy, S. M.) 3. An aspect of local property of absolute summability of the conjugate series of the derived Fourier series. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 18 (1970) 461-465. 4. On the summation of the conjugate series of the derived Fourier series by Borel’s integral method. Univ. Allahabad Studies (N.S.) 2 (1970) no. 3, 191-199.

Floyd, E. E. 02 • 1. Stiefel-Whitney numbers of quaternionic and related manifolds. Trans. Amer. Math. Soc. 155 (1971) 77-94. 2. Gordon Whyburn 1904-1969. Bull. Amer. Math. Soc. 77 (1971) 57-72. (with Jones, F. B.) 282 CHAPTER 9

Andrews, J. J. 03 • 1. On higher-dimensional ﬁbered knots. Trans. Amer. Math. Soc. 153 (1971) 416-426.(with Summers, D. W.)

Rubin, Leonard 04 • 1. Recognizing certain factors of E4. Proc. Amer. Math. Soc. 26 (1970) 199-200. 2. The product of any dogbone space with a line is E4. Duke Math. J. 37 (1970) 189-192.

Husch, L. S. 04 • 1. On piecewise linear unknotting of polyhedra. Yokohama Math. J. 170 (1969) 87-92. 2. Finding a boundary for a 3-manifold. Ann. Math. (2) 91 (1970) 223-235.(with Price, T. M.) 3. Restrictions of isotopies and concordances. Mich. Math. J. 16 (1969) 303-307.(with Rushing, T. R.) 4. Suspension of PL-embeddings. Bull. London Math. Soc. 2 (1970) 191-195. 5. Homotopy groups of PL-embedding spaces. Paciﬁc J. Math. 33 (1970) 149-155.

Tindell R. S. 04 • 1. Classifying bounded 2-manifolds in S4. Osaka J. Math. 7 (1970) 173-177. 2. Relative concordance. Topology of Manifolds (Proc. Inst. Univ. of Georgia, Athens, Georgia, 1969) Markham, Chicago, Illinois, 1970.

Cain, George L. 03 • 1. Mappings with prescribed singular sets. Nieuw Arch. Wisk. (3) 17 (1969) 200-203.

Fuller Richard 03 • 1. Semi-uniform spaces and topological homeomorphism groups. Proc. Amer. Math. Soc. 26 (1970) 365-368. ACADEMIC DESCENDANTS AND PUBLICATIONS 283

Williams, R. F. 02 • 1. The “DA” maps of Smale and structural stability. Global analysis. (Proc. Sympos. Pure Math. Vol. XIV, Berkeley, 1968) 329-334. American Mathematical Society, Providence, Rhode Island, 1970.

Duda, Edwin 02 • 1. Open mappings on spheres. J. Austral. Math. Soc. 10 (1969) 330-334. 2. Compactness of monotone mappings. Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Bingham- ton, Binghamton, N. Y., 1970) 55-77. State University of New York at Binghamton, Binghamton, New York, 1971. 3. Finite-to-one open mappings. Canadian J. Math. 23 (1971) 77- 83. (with Haynsworth, W. H.)

Keesling, J. E. 03 • 1. Compactiﬁcation and the continuum hypothesis. Fund. Math. 66 (1969/70) 53-54. 2. Closed mappings and local dimension. Colloq. Math. 21 (1970) 75-79. 3. Open mappings and closed subsets of the domain in general metric spaces. Proc. Amer. Math. Soc. 20 (1969) 238-245. 4. On certain ideals of C(X). Duke Math. J. 38 (1971) 259-263. (with Nanzetta, Philip) 5. Locally compact full homeomorphism groups are zero-dimensional. Proc. Amer. Math. Soc. 29 (1971) 390-396.

Haynsworth, W. H. 03 • 1. Finite-to-one open mappings. Canadian J. Math. 23 (1971) 77- 83. (with Duda, E.)

Williams, G. K. 02 • 1. Continuous and proper decompositions. Proc. Amer. Math. Soc. 28 (1971) 267-270.

Duke R. A. 02 • 284 CHAPTER 9

1. Geometric embedding of complexes. Amer. Math. Monthly 77 (1970) 597-603.

Dickman, R. F. Jr. 02 • 1. Functionally compact spaces. Paciﬁc J. Math. 31 (1969) 303- 311. (with Zame A.) 2. On closed extensions of functions. Proc. Nat. Acad. Sci. U. S. A. 62 (1969) 326-332.

Garcia-Maynez, A. C. 02 • 1. Some results in the theory of partially continuous functions. Bol. Soc. Mat. Mexicana (2) 1969 1-8. 2. The separation theorem for quasi-closed sets. Proc. Amer. Math. Soc. 27 (1971) 399-404.(with Hunt, J. H. V.)

McMillan, Evelyn 02 • 1. On continuity conditions for functions. Paciﬁc J. Math. 32 (1970) 479-494. ACADEMIC DESCENDANTS AND PUBLICATIONS 285 R. L. Roberts and his Mathematical Descendandts

01 Roberts, J. H. 02 Gilbert, Wilner Paul, Duke University, 1940 03 Zaccaro, Luke N., Syracuse University, 1957 03 Lewis, Jesse C., Syracuse University, 1966 03 Carrano, Frank M., Jr., Syracuse University, 1969 02 Martin, Abram Venable, Jr., Duke University, 1940 02 Civin, Paul, Duke University, 1942 03 Young, Frederick Harris, University of Oregon, 1951 04 Boles, John Amos, Oregon State University, 1968 03 Livingston, Arthur Eugene, University of Oregon, 1952 03 Chrestenson, Hubert Edwin, University of Oregon, 1953 03 Selfridge, Ralph Gordon, University of Oregon, 1953 04 Williams, Louis, University of Florida, 1970 04 Wright, Reverdy, University of Florida, 1971 03 Hunter, Larry Clifton, University of Oregon, 1957 03 Peterson, Donald Palmer, University of Oregon, 1957 03 Dillon, Richard Thomas, University of Oregon, 195 03 Cuttle, Percy Mortimer, University of Oregon, 1959 03 Cuttle, Yvonne Germaine M. C., University of Oregon, 1959 03 Chicks, Charles, University of Oregon, 1960 03 Stafney, James D., University of Oregon, 1963 03 Lindahl, Robert J., University of Oregon, 1964 04 Dungello, Frank, Pennsylvania State University, 1970 03 Bachelis, Gregory Frank, University of Oregon, 1966 03 Iltis, Donald Richard, University of Oregon, 1966 03 White, Christopher Clarke, University of Oregon, 1967 03 Honerlah, Raymond W., University of Oregon, 1968 03 Tomlinson, Michael B., University of Oregon, 1968 02 Hahn, Samuel Wilfred, Duke University, 1948 02 Gentry, Ivey Clinton, Duke University, 1949 02 Jarnagin, Milton Preston, Jr., Duke University, 1950 02 Fulton, Lewis McLeod, Jr., Duke University, 1950 02 Sharp, Henry, Jr., Duke University, 1952 02 Smythe, William R., Duke University, 1955. 02 Gropen, Arthur L., Duke University, 1958 02 Forge, Auguste, Duke University, 1959 02 Kwak, Nosup, Duke University, 1959 286 CHAPTER 9

02 Saadaldin, M. Jawad, Duke University, 1960 02 Hodel, Richard Earl, Duke University, 1962 03 Lobb, Barry Lee, Duke University, 1969 03 Kramer, Thomas Rollin, Duke University, 1971 02 King, Lunsford Richardson, Duke University, 1963 02 Rosenstein, George Morris, Duke University, 1963 03 Bernard, Anthony Dwight, Jr., Case Western Reserve Univ., 1968 02 Wenner, Bruce Richard, Duke University, 1964 02 Vaughan, Jerry E., Duke University, 1965 02 Slaughter, Frank Gill, Duke University, 1966 02 Wilkinson, James, Duke University, 1966 02 Smith, James Clarence, Jr., Duke University, 1967 02 Soniat, Leonard Edward, Duke University, 1967 02 Bookhout, Glen Allen, Duke University, 1970 02 Nichols, Joseph Caldwell, Duke University, 1970 ACADEMIC DESCENDANTS AND PUBLICATIONS 287 Publications of J. H. Roberts and his Mathematical Descendants

Roberts, J. H. 01 • 1. On a problem of C. Kuratowski. Fund. Math. 14 (1929) 92-102. 2. On a problem of G. T. Whyburn. Fund. Math. 13 (1929) 58- 61.(with Dorroh, J. L.) 3. On a problem of Menger concerning regular curves. Fund. Math. 14 (1929) 327-333. 4. A note concerning cactoids. Bull. Amer. Math. Soc. 36 (1930) 894896. 5. Concerning atroidic continua. Monat. Mat. Physik. 37 (1930) 223-230. 6. Concerning collections of continua not all bounded. Amer. J. Math. 52 (1930) 551-562. 7. Concerning non-dense plane continua. Trans. Amer. Math. Soc. 32 (1930) 6-30. 8. A non-dense plane continuum. Bull. Amer. Math. Soc. 37 (1931) 720-722. 9. A point set characterization of closed two-dimensional manifolds. Fund. Math. 18 (1931) 39-46. 10. Concerning metric collections of continua. Amer. J. Math. 53 (1931) 422-426. 11. Concerning topological transformations in En . Trans. Amer. Math. Soc. 34 (1932) 252-262. 12. Concerning uniordered spaces. Proc. Nat. Aca. Sci. 18 (1932) 403406. 13. A property related to completeness. Bull. Amer. Math. Soc. 38 (1932) 835-838. 14. Concerning compact continua in certain spaces of R. L. Moore. Bull. Amer. Math. Soc. 39 (1933) 615-621. 15. On a problem of Knaster and Larankiewicz. Bull. Amer. Math. Soc. 40 (1934) 281-283. 16. Collections ﬁlling a plane. Duke Math. J. 2 (1936) 10-19. 17. Note on topological mappings. Duke Math. J. 5 (1939) 428-430. 18. Two-to-one transformations. Duke Math. J. 6 (lg40) 256-262. 288 CHAPTER 9

19. A theorem on dimension. Duke Math. J. 8 (1941) 565-574. 20. Open transformati:ons and dimension. Bull. Amer. Math. Soc. 53 (1947 ) 176-178 . 21. A problem in dimension theory. Amer. J. Math. 70 (1948) 126-128. 22. A nonconvergent iterative process. Proc. Amer. Math. Soc. 4 (1953) 640-644 . 23. The rational points in Hilbert space. Duke Math. J. 23 (1956) 489-492 . 24. Problem of Treybig concerning separable spaces. Duke Math. J. 28 153-156 (1961) 25. Contractibility in spaces of homeomorphisms. Duke Math. J. 28 (1961) 213-220. 26. Solution to Aufgabe 260 (second part) 1 ) . Elem. Math. 16 (1961) 109-111. 27. Concerning some problems raised by A. Lelek. Fund. Math. 54 (1964) 325-334. (with King, L. R. and Rosenstein, G. M. , Jr.) 28. Zero-dimensional sets blocking connectivity functions. Fund. Math. 57 (1965) 173-179. 29. Realizability of metric-dependent dimensions. Proc. Amer. Math. Soc. 18 (1968) 1439-1442.

30. Metric-dependent function d2 and covering dimension. Duke Math. J. 38 (1970) 467-472. 31. Sections of continuous collections. Bull. Amer. Math. Soc. 49 (1943) 142-143.(with Civin, Paul) 32. On a certain nonlinear integral equation of the Volterra type. Paciﬁc J. Math. 1 (1951) 431-445.(with Mann, W. R.) 33. Two-to-one transformations on two-manifolds. Trans. Amer. Math. Soc. 49 (1941) 1-17.(with Martin, A. V) 34. A note on countable-dimensional spaces. Proc. Japan Academy 41 (1965) 155-158.(with Nagami, Keio) 35. Metric-dependent dimension functions. Proc. Amer. Math. Soc. 16 (1965) 601-604.(with Nagami, Keio) 36. A study of metric dependent dimension functions. Trans. Amer. Math. Soc. 129 (1967) 414-435.(with Nagami, Keio) 37. Metric dimension and equivalent metrics. Fund. Math. 62 (1968) 1-5.(with Slaughter, F. G.) ACADEMIC DESCENDANTS AND PUBLICATIONS 289

38. Characterization of dimension in terms of the existence of a continuum. Duke Math. J. 37 (1970) 681-688.(with Slaughter, F. G.) 39. Monotone transformations of 2-dimensional manifolds. Annals of Math. 39 (1938) 851-862.(with Steenrod, N. E.)

Gilbert, Paul W. 02 • 1. n to one mappings of linear graphs. Duke Math. J. 9 (1942) 475-486.

Martin, Abram V., Jr. 02 • 1. Decompositions and quasi-compact mappings. Duke Math. J. 21 (1954) 463-469. 2. A note on derivatives and neighborly functions. Proc. Amer. Math. Soc. 8 (1957) 465-467. 3. On a method of Courant for minimizing functionals. J. Math. and Phys. 41 (1962) 291-299.(with Butler, Terence)

Civin Paul 02 • 1. Inequalities for trigonometric integrals. Duke Math. J. 8 (1941) 656-665. 2. Two-to-one mappings of manifolds. Duke Math. J. 10 (1943) 49-57. 3. Polynomial dominants. Bull. Amer. Math. Soc. 52 (1946) 352-356. 4. Fourier coefficients of dominant functions. Duke Math. J. 13 (1946) 1-7. 5. Mean values of periodic functions. Bull. Amer. Math. Soc. 53 (1947) 530-535. 6. Approximation in Lip (,p). Bull. Amer. Math. Soc. 55 (1949 794-796. 7. Approximation to conjugate functions. Proc. Amer. Math. Soc. 2 (1951) 207-208. 8. Multiplicative closure in the Walsh functions. Pacific J. Math. 2 (1952) 291-295. 9. Orthonormal cyclic groups. Pacific J. Math. 4 (1954) 481-482. 290 CHAPTER 9

10. Abstract Riemann Sums. Pacific J. Math. 5 (1955) 861-868. 11. Some ergodic theorems involving two operators. Pacific J. Math. 5 (1955) 869-876. 12. Correction to ‘Some erogodic theorems involving two operators.’ Pacific J. Math. 6 (1956) 795. 13. A maximum modulus property of maximal subalgebras. Proc. Amer. Math. Soc. 10 (1959) 51-54. 14. Involutions on locally compact rings. Pacific J. Math. 10 (1960) 1199-1202. 15. Isometries of group algebras. Proc. Amer. Math. Soc. 11 (1960) 983-985. 16. Extensions of homomorphisms. Pacific J. Math. 11 (1961) 1223- 1233. 17. Ideals in the second conjugate algebra of a group algebra. Math. Scand. 11 (1962) 171-174. 18. Annihilators inthe second conjugate algebra of a group algebra. Pacific J. Math. 12 (1962) 855-862. 19. Weak structural synthesis for certain Banach algebras. Trans. Amer. Math. Soc. 104 (1962) 420-424. 20. The Multiplicity of a class of perfect sets. Proc. Amer. Math. Soc. 4 (1953) 260-263.(with Chrestenson, H. E.) 21. Sections of continuous collections. Bull. Amer. Math. Soc. 49 (1943) 142-143.(with Roberts, J. H.) 22. Maximal closed preprimes in Banach algebras. Trans. Amer. Math. Soc. 147 (1970J 241-260.(with White, C. C.) 23. Ideals in multiplicative semi-groups of continuous functions. Duke Math. J. 23 (1956) 325-334.(with Yood, Bertram) 24. Ideals in multiplicative semi-groups of continuous functions, A correction. Duke Math. J. 23 (1956) 631. 25. Invariant functionals. Pacific J. Math. 6 (1956) 231-237. 26. Regular Banach algebras with a countable space of maximal regular ideals. Proc. Amer. Math. Soc. 7 (1956) 1005-1010. 27. Quasi-reflexive spaces. Proc. Amer. Math. Soc. 8 (1957) 906911 . 28. Involutions on Banach algebras. Pacific J. Math. 9 (1959) 415436.(with Yood, Bertram) ACADEMIC DESCENDANTS AND PUBLICATIONS 291

29. The second conjugate space of a Banach algebra as an algebra. Paciﬁc J. Math. 11 (1961) 847-870. 30. Lie and Jordan structures in Banach algebras. Paciﬁc J. Math. 15 (1965) 775-797.(with Yood, Bertram)

Young, F. H. 03 • 1. A note on summation. Amer. Math. Monthly 57 (1950) 625. 2. Transformations of Fourier coeﬃcients. Proc. Amer. Math. Soc. 3 (1952) 783-791. 3. The NOTS REAC. Amer. Math. Monthly 60 (1953) 237-243. 4. Analysis of shift register counters. J. Assoc. Comput. Mach. 5 (1958) 385-388.

Livinston, Arthur E. 03 • 1. Some Hausdorff means which exhibit the Gibhs’ phenomenon. Pacific J. Math. 3 (1953) 407-415. 2. The Space HP, O p 1, is not normable. Pacific J. Math. 3 (1953) 613-616. 3. A necessary condition for the convergence of f(x)dx. Amer. Math. Monthly 61 (1954) 250-251. 4. The Lebesgue constants for Euler (E, p) summation of Fourier series. Duke Math. J. 21 (1954) 309-313. 5. A generalization of an inequality due to Beurling. Pacific J. Math. 4 (1954) 251-257. 6. The zeros of a certain class of indefinite integrals. Proc. Amer. Math. Soc. 5 (1954) 296-300. 7. The series f(n)/n for periodic f. Canad. Math. Bull. 8 X (1965) 413-432. 8. Meromorphic multivalent close-to-convex functions. Trans. Amer. Math. Soc. 119 (1965) 167-177. 9. On the radius of univalence of certain analytic functions. Proc. Amer. Math. Soc. 17 (1966) 352-357. 10. A theorem on duality mappings in Banach spaces. Ark. Math. 4 (1962) 405-411.(with Beurling, Arne) 11. The zeros of certain sine-like integrals. Proc. Amer. Math. Soc. 7 (1956) 813-816.(with Lorch, Lee) 292 CHAPTER 9

12. The zeros of some oscillating integrals. J. Math. Anal. Appl. 2 (1961) 438-445.(with Lorch, Lee) 13. A short proof of a classical theorem in the theory of Fourier integrals. Amer. Math. Monthly 62 (1955) 434-437.(with Riesz, Marcel)

Chrestenson, Hubert E. 03 • 1. A class of generalized Walsh functions. Paciﬁc J. Math. 5 (1955) 17-31. , and Civin, Paul 2. The multiplicity of a class of perfect sets. Proc. Amer. Math. Soc. 4 (1953) 260-263.

Selfridge, R. G. 03 • 1. Approximations with least maximum error. Pacific J. Math. 3 (1953) 247-255. 2. Generalized Walsh transformations. Pacific J. Math. 5 (1955) 451480. 3. A table of the incomplete elliptic integral of the third kind. Dover Publications, Inc., New York, (1958) xiv+805. 4. On a form representing a large collection of primes. J. Natur. Sci. and Math. 6 (1966) 241-242.(with Chawla, L. M., Maxfield, John E.) 5. The surface area of a screw. Appl. Sci. Res. 8 (1959) 377385.(with Maxfield, John E.) 6. Similarity classifications of complex matrices. Math. Mag. 34 (1960/61) 147-152.(with Maxfield, John E.)

Hunter, Larry C. 03 • 1. On induced topologies in quasi-reﬂexive Banach spaces. Proc. Amer. Math. Soc. 11 (1960) 161-163. 2. Optimum checking procedures. Statistical Theory of Reliability (Proc. Advanced Seminar, Math. Res. Center, U. S. Army, Univ. of Wisconsin, Madison, Wis., 1962) Univ. of Wisconsin Press, Madison, Wis., 1963, 95-113. 3. Asymptotic solutions of certain linear diﬀerence equations with applications to some eigenvalue problems. J. Math. Anal. Appl. 24 (1968) 279-289. ACADEMIC DESCENDANTS AND PUBLICATIONS 293

4. Optimum redundancy when components are subject to two kinds of failure. J. Soc. Indust. Appl. Math. 11 (1963) 64-73. (with Barlow, Richard E.; Proschan, Frank) 5. Optimum checking procedures. J. Soc. Indust. Appl. Math. 11 (1963) 1078-1095.

Peterson, D. P. 03 • 1. Linear interpolation, extrapolation, and prediction of random space-time ﬁelds with a limited domain of measurement. IEEE Trans., Information Theory IT-ll (1965) 18-30.(with Middleton, D. P.)

Cuttle, P. M. 03 • 1. Operators on a quasi-reﬂexive Banach space. Proc. Amer. Math. Soc. 13 (1962) 993-997.

Cuttle, Yvonne 03 • 1. On quasi-reﬂexive Banach spaces. Proc. Amer. Math. Soc. 12 (1961) 936-940.

Stafney, James D. 03 • 1. Arens multiplication and convolution. Paciﬁc J. Math. 14 (1964) 1423-1447. 2. A permissible restriction on the coeﬃcients in uniform polynomial approximation to C[0, 1]. Duke Math. J. 34 (1967) 393-396. 3. An unbounded inverse property in the algebra of absolutely convergent Fourier series. Proc. Amer. Math. Soc. 18 (1967) 497-498. 4. Approximation of Wp-continuity sets by p-sideon sets. Mich. Math. J. 16 (1969) 161-176. 5. The spectrum of an operator on an interpolation space. Trans. Amer. Math. Soc. 144 (1969) 333-349.

Bachelis, Gregory F. 03 • 1. Homomorphisms of annihilator Banach algebras. Paciﬁc J. Math. 25 (1968) 229-247. 294 CHAPTER 9

2. Homomorphisms of annihilator Banach algebras. II. Paciﬁc J. Math. 30 (1969) 283-291.

Iltis, Richard 03 • 1. Some algebraic structure in the dual of a compact group. Canad. J. Math. 20 (1968) 1499-1510.

White, C. C. 03 • 1. Maximal closed preprimes in Banach algebras. Trans. Amer. Math. Soc. 147 (1970) 241-260.(with Civin, Paul)

Hahn, Samual W. 02 • 1. Universal spaces under strong homeomorphisms. Trans. Amer. Math. Soc. 70 (1951) 301-311.

Gentry, Ivey C. 02 • 1. On the characteristic roots of tournament matrices. Bull. Amer. Math. Soc. 74 (1968) 1133-1135.(with Brauer, Alfred) 2. A new proof of a theorem by H. G. Landau on tournament matrices. J. Combinatorial Theory 5 (1968) 289-292.(with Brauer, Alfred; Shaw, Kay)

Jarnagin, M. P., Jr. 02 • 1. Integration of the general bivariate Gaussian distribution over an oﬀset circle. Math. Comp. 15 (1961) 375-382.(with DiDonato, A. R.) 2. A method for computing the circular coverage function. Math. Comp. 16 (1962) 347-355. 3. The eﬃcient calculation of the incomplete Beta-function ratio for half integer values of the parameters a,b. Math. Comp. 21 (1967) 652-662.

Fulton, Lewis M., Jr. 02 • 1. Decompositions induced under ﬁnite-to-one closed mappings. Duke Math. J. 18 (1951) 287-295.

Sharp, Henry S., Jr. 02 • ACADEMIC DESCENDANTS AND PUBLICATIONS 295

1. A comparison of methods for evaluating the complex roots of quartic equations. J. Math. Phys. Mass. Inst. Tech. 20 (1941) 243258. 2. Strongly topological imbedding of F6-subsets of E . Amer. J. Math. 75 (1953) 557-564. 3. Autohomeomorphisms on E . Fund. Math. 59 (1966) 171-175. 4. Denseness and completeness in certain function spaces. Amer. Math. Monthly 74 (1967) 266-271. 5. Quasi-orderings and topologies on ﬁnite sets. Proc. Amer. Math. Soc. 17 (1966) 1344-1349. 6. Cardinality of ﬁnite topologies. J. Comb. Theory 5 (1968) 82-86.

Smythe, William R., Jr. 02 • 1. A theorem on upper-semicontinuous decompositions. Duke Math. J. 22 (1955) 485-495. 2. Charged sphere in cylinder. J. Appl. Phys. 31 (1960) 553-556. 3. Flow around a sphere in a circular tube. Phys. Fluids 4 (1961) 756-759. 4. Charged sphereoid in cylinder. J. Math. Phys. 4 (1963) 833-837. 5. Introduction to linear programming with applications. Prentice Hall, Inc., Englewood Cliﬀs, N. J., (1966) viii+221. (with John- son, Lynwood A.)

Gropen Arthur L. 02 •

1. Special homeomorphisms in the functional space C(X,I2n+1) Duke Math. J. 28 (1961) 629-637.

Forge, Auguste 02 • 1. Dimension preserving compactiﬁcations allowing extensions of continuous functions. Duke Math. J. 12 (1964) 1-5.

Saadaldin, M. J. 02 • 1. A generalized Lebesgue covering theorem. Duke Math. J. 29 (1962) 539-542.

Hodel, Richard Earl 02 • 296 CHAPTER 9

1. Open functions and dimension. Duke Math. J. 30 (1963) 461- 467. 2. Total normality and the hereditary property. Proc. Amer. Math. Soc. 17 (1966) 462-465. 3. Note on metric-dependent dimension functions. Fund. Math. 61 (1967) 83-89. 4. Sum theorems for topological spaces. Paciﬁc J. Math. 30 (1969) 59-65.

King, L. R. 02 • 1. Concerning some problems raised by A. Lelek. Fund. Math. 54 (1964) 325-334.(with Roberts, J. H.; Rosenstein, G. M., Jr.)

Rosenstein G. M. 02

1. A further extension of Lebesgue’s covering theorem. Proc. Amer. Math. Soc. 15 (1964) 683-788. 2. Concerning some problems raised by A. Lelek. Fund. Math. 54 (1964) 325-334.(with King, L. R.; Roberts, J. H.)

Wenner, B. R. 02 • 1. Dimension on boundaries of E-spheres. Paciﬁc J. Math. (1968) 201210. 2. Remetrization in strongly countable-dimensional spaces. Canad. J. Math. 21 (1969) 748-750. 3. Note on a theorem on J. Nagata. Compositio Math. 21 (1969) 4-6.(with Vaughan, J. E.)

Vau han J E 02 • 1. A modiﬁcation of Morita’s characterization of dimension. Paciﬁc J. Math. 20 (1967) 189-196. 2. Linearly ordered collections and paracompactness. Proc. Amer. Math. Soc. 24 (1970) 168-172 3. Note on a theorem by J. Nagata. Compositio Math. 21 (1969) 4-6.(with Wenner, B. R.)

Slaughter, Frank Gill, Jr. 02 • ACADEMIC DESCENDANTS AND PUBLICATIONS 297

1. A note on inverse images of closed mappings. Proc. Japan Acad. 44 (1968) 628-632. 2. Metric dimension and equivalent metrics. Fund. Math. 62 (1968) 1-5.(with Roberts, J. H.)

Wilkinson James B. 02 • 1. A lower bound for the dimension of certain B sets in completely normal spaces. Proc. Amer. Math. Soc. 20 (1969) 175-178.

Smith, James C., Jr. 02 • 1. Characterizations of metric-dependent dimension functions. Proc. Amer. Math. Soc. 19 (1968) 1264-1269. 2. Lebesgue characterizations of uniformity-dimension functions. Proc. Amer. Math. Soc. 22 (1969) 164-169.

Soniat, Leonard E. 02 • 1. A new characterizaation of Hausdorﬀ K-spaces. Proc. Japan Acad. 44 (1968) 1031-1032.(with Lin, Y. F.)

Nichols, J. C. 02 • 1. Equivalent metrics giving diﬀerent values to metric-dependent dimension functions. Proc. Amer. Math. Soc. 23 (1969) 648- 652.

Roberts, J. H. 01

1. Metric-dependent function d2 and covering dimension. Duke Math. J. 37 (1970) 467-472. 2. Characterization of dimension in terms of the existence of a continuum. Duke Math. J. 37 (1970) 681-688.(with Slaughter, F. G.)

Bachelis, G. F. 03 •

1. On the ideal of unconditionally convergent Fourier series in Lp(G). Proc. Amer. Math. Soc. 27 (1971) 309-312.

Sharp, Henry 02 • 298 CHAPTER 9

1. The permanent of a transitive relation. Proc. Amer. Math. Soc. 26 (1970) 153-157 .

Hodel, R. E. 02 • 1. A note on subparacompact spaces. Proc. Amer. Math. Soc. 25 (1970) 842-845 .

Wenner, B. R. 02 • 1. Sums of ﬁnite-dimensional spaces. Bull. Austral. Math. Soc. (1969) 357-361. 2. Extending maps and dimension theory. Duke Math. J. 37 (1970) 627-631 . 3. Dimensions-theoretic properties of completions. Proc. Amer. Math. Soc. 28 (1971) 590-594.

Vaughan, J. E. 02 • 1. Spaces of countable and point-countable type. Trans. Amer. Math. Soc. 151 (1970) 341-351. 2. Perfect mappings and spaces of countable type. Canadian J. Math. 22 (1970) 1208-1210. 3. Paracompactness and elastic spaces. Proc. Amer. Math. Soc. 28 (1971) 299-303 .(with Tamano, Hisahiro)

Slaughter, F. G. 02 • 1. Characterization of dimension in terms of the existence of a continuum. Duke Math. J. 37 (1970) 681-688. (with Roberts, J. H.)

Bookhout, G. A. 02 • 1. Metric dimension of complete metric spaces. Proc. Amer. Math. Soc. 24 (1970) 754-759.

Cleveland, C. M. 01 • 1. Concerning points of a continuous curve that are not accessible from each other. Proc. Nat. Acad. Sci. 18 (1927) 275-276. ACADEMIC DESCENDANTS AND PUBLICATIONS 299

2. On the existence of acyclic curves satisfying certain conditions with respect to a given continuous curve. Trans. Amer. Math. Soc. 33 (1931) 958-978.

Dorroh, J. L. 01 • 1. Concerning a set of metrical hypothesis for geometry. Ann. of Math. (2) 29 (1928). 2. On a problem of G. T. Whyburn. Fund. Math. (1929) (with Roberts, J. H.) 3. Concerning a set of axioms for the semi-quadratic geometry of a three-space. Bull. Amer. Math. Soc. 36 (1930). 4. Concerning adjunctions to algebras. Bull. Amer. Math. Soc. 38 (1932). 5. Concerning the direct product of algebras. Ann. of Math. 36 (1935). 6. On the rational roots of polynomial equations. Amer. Math. Monthly 53 (1946) 383-384.(with Howell, L. B.) 7. Some metric propoerties of descriptive planes. Amer. J. Math. 43 (1931).

Vickery, C. W. 01 • 1. Spaces in which there exist uncountable convergent sequences of points. Tahoku Math. J. 40 (1935) 1-26. 2. Spaces of uncountably many dimensions. Bull. Amer. Math. Soc. 45 (1939) 456-462. 3. Axioms for Moore spaces and metric spaces. Bull. Amer. Math. Soc. 46 (1940) 560-564. 4. Cyclically invariant graduation. Econometrica. 12 (1944) 19-25.

Klipple, E. C. 01 • 1. Two-dimensional spaces in which there exist contiguous points. Trans. Amer. Math. Soc. 44 (1938) 250-276.

Basye, R. E. 01 • 1. Concerning two internal properties of plane continua. Bull. Amer. Math. Soc. 41 (1935) 670-674. 300 CHAPTER 9

2. Simply connected sets. Trans. Amer. Math. Soc. 38 (1935) 341356. 3. Some separation properties of the plane. Amer. J. Math. 58 (1936) 323-328. ACADEMIC DESCENDANTS AND PUBLICATIONS 301 F. Burton Jones and his Mathematical Descendandts

01 Jones, F. Burton 02 McAuley, L. F., University ofNorth Carolina, 1954 03 McAllister, Byron L., University of Wisconsin, 1966 03 Ungar, Gerald S., Rutgers University, 1967 03 Wilson, David, Rutgers University, 1969 03 Christoph, Francis T., Rutgers University, 1969 03 Addis, David, Rutgers University, 1970 03 Haver, William E., SUNY, Binghamton, 1970 03 Reed, Myra S. SUNY, Binghamton, 1971 03 Baildon, John D. SUNY, Binghamton, 1971 03 Woodruﬀ, Edythe P., SUNY, Binghamton, 1971 02 Smith, Marion B., University of North Carolina, 1957 02 Grace, E. E., University of North Carolina, 1957 03 Hagopian, Charles L., Arizona State University, 1968 03 Berg, Gordon Owen, Arizona State University, 1969 03 Fitzgerald, Robert W.j Arizona State University, 1969 03 Schlais, Harold Eugene, Arizona State University, 1971 02 Heath, Robert W., Univerity of North Carolina, 1959 02 Roy, Prabir, University of North Carolina, 1962 02 Thomas, Edward Sandusky, Jr., University of California, Riverside, 1965 02 Greef, Lynn George, University of California, Riverside, 1966 02 Arnquist, Cliﬀord W., University of California, Riverside, 1967 02 Vought,Eldon Jon, University of California, Riverside, 1967 02 Rogers, James Ted, Jr., University of California, Riverside, 1967 02 Shirley, Edward D., University of California, Riverside, 1969 02 Rogers Leland Edward, University of California, Riverside, 1970 02 Gordh, George Rudolph, University of California, Riverside, 1971 302 CHAPTER 9 Publications of F. Burton Jones and his Mathematical Descendants

Jones, F. Burton 01 • 1. A theorem concerning locally peripherally separable spaces. Bull. Amer. Math. Soc. 41 (1935) 437-439. 2. Concerning certain topologically ﬂat spaces. Trans. Amer. Math. Soc. 42 (1937) 53-93. 3. Concerning normal and completely normal spaces. Bull. Amer. Math. Soc. 43 (1937) 671-677. 4. Concerning R. L. Moore’s Axiom 5. Bull. Amer. Math. Soc. 44 (1938) 689-692. 5. Concerning the boundary of a complementary domain of a continuous curve. Bull. Amer. Math. Soc. 45 (1939) 428-435. 6. Concerning certain linear abstract spaces and simple continuous curves. Bull. Amer. Math. Soc. 45 (1939) 623-628. 7. Certain equivalences and subsets of the plane. Duke Math. J. 5 (1939) 133-145. 8. Almost cyclic elements and simple links of a continuous curve. Bull. Amer. Math. Soc. 46 (1940) 661-664. 9. Aposyndetic continua and certain boundary value problems. Amer. J. Math. 63 (1941) 544-545. 10. Connected and disconnected plane sets and the functional equation f(x) + f(y) = f(x + y). Bull. Amer. Math. Soc. 48 (1942) 115120. 11. Measure and other properties of a Hamel basis. Bull. Amer. Math. Soc. 48 (1942) 472-481. 12. Concerning the separability of certain locally connected metric spaces. Bull. Amer. Math. Soc. 52 (1946) 303-306. 13. A characterization of a semi-locally connected plane continuum. Bull. Amer. Math. Soc. 52 (1947) 170-175. 14. Concerning non-aposyndetic continua. Amer. J. Math. 70 (1948) 403-413. 15. A note on homogeneous plane continua. Bull. Amer. Math. Soc. 55 (1949) 113-114. 16. Certain homogeneous unicoherent indecomposable continua. Proc. Amer. Math. Soc. 2 (1951) 855-859. ACADEMIC DESCENDANTS AND PUBLICATIONS 303

17. Concerning aposyndetic and non-aposyndetic continua. Bull. Amer. Math. Soc. 58 (195Z) 137-151. (This is the published text of his invited hour address given before the American Math- ematical Society, Boulder, Colorado, September 1, 1949.) 18. On the separation of the set of pairs-of a set. J. Elisha Mitchell Sci.Soc. 68 (1952) 44-45. 19. On certain well-ordered monotone collections of sets. J. Elisha Mitchell Sci. Soc. 69 (1953) 30-34. 20. On a property related to separability in metric spaces. J. Elisha Mitchell Sci. Soc. 70 (1954) 30-33. 21. On a certain type of homogeneous plane continuum. Proc. Amer. Math. Soc. 6 (1955) 735-740. 22. On the existence of weak cut points in plane continua. Proc. Amer. Math. Soc. 9 (1958) 530-532. 23. R. L. Moore’s Axiom 1’ and metrization. Proc. Amer. Math. Soc. 9 (1958) 487. 24. Moore spaces and uniform spaces. Proc. Amer. Math. Soc. 9 (1958) 483-486. 25. Product spaces in n-manifolds. Proc. Amer. Math. Soc. 10 (1959) 171-192. 26. On the first axiom of countability for locally compact Hausdorff spaces. Colloq. Math. 7 (1959) 33-34. 27. Another cutpoint theorem for plane continua. Proc. Amer. Math. Soc. 11 (1960) 556-558. 28. The cyclic connectivity of plane continua. Pacific J. Math. 11 (1961) 1013-1016. 29. On the existence of a small connected open set with a connected boundary. Bull. Amer. Math. Soc. 68 (1962) 117-119. 30. A fixed point free mapping of a connected plane set. Colloq. Math. 11 (1963) 73-74. 31. Stone’s 2-sphere conjecture. Amer. J. Math. 87 (1965) 497-501. 32. Remarks on the normal Moore space metrization problem. An- nals of Math. Studies. 60 (1966) 115-119.

33. Connected Gδ-graphs. Duke Math. J. 33 (1966) 341-345. (with Thomas, E. S.) 34. On the plane one-to-one map of a line. Colloq. Math. 19 (1968) 231-233. 304 CHAPTER 9

35. Stronger forms of aposyndetic continua. Topology Conference, Arizona State University (1967) edited by E. E. Grace (1967) 170-173.(with Vought, E. J.) 36. Fake Souslin trees. Duke Math. J. 36 (1969). 37. Countable locally conncected Urysohn spaces.(with Stone, A. H.) 38. One-to-one continuous images of a line. Fund. Math.

McAuley Louis F. 02 • 1. An atomic decomposition of continua into aposyndetic continua. Trans. Amer. Math. Soc. 88 (1958) 1-11. 2. Some upper semi-continuous decompositions of E3 into E3. Ann. of Math. (2) 73 (1961) 437-457. 3. Upper semi-continuous decompositions of E3 into E3 and generalizations of metric spaces. Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice- Hall, Englewood Cliﬀs, N. J., (1962) 21-26. 4. Lifting disks and certain light open mappings. Proc. Nat. Acad. Sci. 53 (1965) 255-260. 5. Concerning a conjecture of Whyburn on light open mappings. Bull. Amer. Math. Soc. 71 (1965) 671-674. 6. Conditions for the equability of the inductive dimensions. Port. Math. 24 (1965) 21-30. 7. Conditions under which light open mappings are homeomorphisms. Duke Math. J. 33 (1966) 445-452. 8. Another decomposition of E3 into points and intervals. Topology Seminar (Wisconsin, 1965) 33-51. Ann. of Math. Studies, No. 60, Princeton University Press, Princeton, N. J., 1966. 9. Completely regular mappings, ﬁber spaces, the weak bundle properties, and the generalized slicing structure properties. Topol- ogy Seminar (Wisconsin, 1965) 219-226. Ann. of Math. Studies, No. 60, Princeton University Press, Princeton, J. J., 1966. 10. 33 The existence of a complete metric for a special mapping space and some consequences. Topology Seminar (Wisconsin, 1965) 135-139. Ann. of Math. Studies, No. 60, Princeton University Press, Princeton, N. J., 1966. ACADEMIC DESCENDANTS AND PUBLICATIONS 305

11. Open mappings and open problems. Topology Conference (Ari- zona State University, Tempe, Arizona, 1967), 184-202, Arizona State University, Tempe, Arizona, 1968. 12. Certain mappings of decompositions which are topologically projections. Bull. Amer. Math. Soc. 75 (1969) 941-944. 13. Whyburn’s conjecture for some diﬀerentiable maps. Proc. Nat. Acad. Sci. U. S. A. 56 (1966) 405-412.(with Cronin-Scanlon, Jane) 14. Semigroups on continua rules by arcs. Fund. Math. 56 (1964) 1-8.(with Koch, R. J.) 15. Semigroups on trees. Fund. Math. 50 (1961/62) 341-346. 16. On hereditarily locally connected spaces and one-to-one continuous images of a line. Colloq. Math. 17 (1967) 319-324.(with Lelek, A.) 17. A new ”cyclic” elementary theory. Math Zeit. 101 (1967) 152164.(with McAllister, B. L.) 18. Fiber spaces and n-regularity. Topology Seminar (Wisconsin, 1965) 33-51. Ann. of Math. Studies, No. 60, Princeton Univer- sity Press, Princeton, N. J., 1966.(with Tulley, P. A.) 19. Lifting cells for certain light open mappings. Math. Ann. 175 (1968) 114-120.(with Tulley, P. A.)

Ungar, Gerald S. 03 • 1. Local homogeneity. Duke Math. J. 34 (1967) 693-700. 2. Light fiber maps. Fund. Math. 62 (1968) 31-45. 3. Pseudoregular mappings. Colloq. Math. 19 (1968) 225-229. 4. A pathological fiber space. Illinois J. Math. 12 (1968) 623-625. 5. Completely regular maps, fiber maps and local n-connectivity. Proc. Amer. Math. Soc. 21 (1969) 104-108. 6. Conditions for a mapping to have the slicing structure property. Pacific J. Math. 30 (1969) 549-553.

Grace, E. E. 02 • 1. A note on linear spaces and unicoherence. J. Elisha Mitchell Sci. Soc. 70 (1954) 33-34. 2. Cut sets in totally nonaposyndetic continua. Proc. Amer. Math. Soc. 9 (1958) 98-104. 306 CHAPTER 9

3. Totally non-connected im kleinen continua. Proc. Amer. Math. Soc. 9 (1958) 808-821. 4. A totally nonaposyndetic, compact, Hausdorff space with no cut point. Proc. Amer. Math. Soc. 15 (1964) 281-283. 5. Cutpoints in totally non-semilocally-connected continua. Pacific J. Math. 14 (1964) 1241-1244. 6. On local properties and G sets. Pacific J. Math. 14 (1964) 12451248. 7. Certain questions related to the equivalence of local connectedness and connectedness im kleinen. Colloq. Math. 13 (1964/65) 211-216. 8. Peripheral covering properties imply covering properties. Canad. J. Math. 20 (1968) 257-263. 9. On the existence of generalized cut points in strongly non-locally connected continua. Proc. Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) 138-146. Arizona State Univ., Tempe, Ariz., 1968) 10. Separability and metrizability in pointwise paracompact Moore spaces. Duke Math. J. 31 (1964) 603-613.(with Heath, R. W.)

Hagopian, Charles L. 03 • 1. On non-aposyndesis and the existence of a certain generalized cut point. Proc. Topology Conference (Arizona State Univ., Tempe, Ariz,, 1967) 327-329 Arizona State Univ., Tempe, Ariz., 1968. 2. Mutual aposyndesis. Proc. Amer. Math. Soc. 23 (1969) 615- 622. 3. Concerning semi-local connectedness and cutting in nonlocally connected continua. Paciﬁc J. Math. 30 (1969) 657-662. 4. Concerning arcwise connectedness and the existence of simple closed curves in plane continua. Trans. Amer. Math. Soc. 147 (1970) 389-402.

Fitzgerald, R. W. 03 • 1. The Cartesian product of non-degenerate compact continua is n-point aposyndetic. Proc. Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968 324-326. ACADEMIC DESCENDANTS AND PUBLICATIONS 307

2. Core decompositions of continua. Fund. Math. 61 (1967) 33-50. (with Swingle, P. M.)

Heath, Robert W. 02 • 1. A regular semi-metric space for which there is no semi-metric under which all spheres are open. Proc. Amer. Math. Soc. 12 (1961) 810-811. 2. Arc-wise connectedness in semi-metric spaces. Paciﬁc J. Math. 12 (1962) 1301-1319. 3. Screenability, pointwise paracompactness, and metrization of Moore spaces. Canadian J. Math. 16 (1964) 763-770.

4. Separability and X1-compactness. Colloq. Math. 12 (1964) 11- 14. 5. On open mappings and certain spaces satisfying the ﬁrst countability axiom. Fund. Math. 57 (1965) 91-96. 6. On spaces with point-countable bases. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 393-395. 7. A paracompact semi-metric space which is not an M3-space. Proc. Amer. Math. Soc. 17 (1966) 868-870. 8. Topological well-ordering and continuous selections. Invent. Math. 6 (1968) 150-158.(with Engelking, J. R.; Michael, E.) 9. Separability and metrizability in pointwise paracompact Moore spaces. Duke Math. J. 31 (1964) 603-610.(with Grace, E. E.)

Roy, Prabir 02 • 1. Failure of equivalence of dimension concepts for metric spaces. Bull. Amer. Math. Soc. 68 (1962) 609-613. 2. On the properties and construction of HGD designs with m- associate classes. Calcutta Statist. Assoc. Bull. 11 (1962) 10-38. 3. Separability of connected, locally connected, metric spaces. Duke Math. J. 29 (1962) 99-100. 4. A countable connected Urysohn space with a dispersion point. Duke Math. J. 33 (1966) 331-333. 5. Nonequality of dimensions for metric spaces. Trans. Amer. Math. Soc. 134 (1968) 117-132. A Holder type inequality for symmetric matrices with non-negative entries. Proc. Amer. Math. Soc. 16 (1965) 1244-1245. (with Blakley, G. R.) 308 CHAPTER 9

Thomas, Edward S., Jr. 02 • 1. A stronger bridge theorem. Colloq. Math. 13 (1964/65) 209-210. 2. Some characterizations of functions of Baire class 1. Proc. Amer. Math. Soc. 17 (1966) 456-461. 3. Monotone decompositions of irreducible continua. Rozprawy Math. 50 (1966) 74. 4. Spaces determined by their homeomorphism groups. Trans. Amer. Math. Soc. 126 (1967) 244-250. 5. A classification of continua by certain cutting properties. Fund. Math. 60 (1967) 139-148. 6. Extending real maps defined on a subset of a disk. Proc. Amer. Math. Soc. 20 (1969) 75-80.(with Ball, B. J.; Ford, Jo) 7. Aligning functions defined on Cantor sets. Trans. Amer. Math. Soc. 141 (1969) 63-69.(with Ford, Jo) 8. Mazur’s theorem on sequentually continuous linear functionals. Proc. Amer. Math. Soc. 14 (1963) 644-647.(with Isbell, John R.) 9. Connected G graphs. Duke Math. J. 33 (1966) 341-345. (with Jones, F. Burton) 10. Isomorphic cone-complexes. Pacific J. Math. 22 (1967) 345-348. (with Segal, Jack)

Vought, Eldon J. 02 • 1. A classiﬁcation scheme and characterization of certain curves. Colloq. Math. 20 (1969) 91-98. 2. A characterization of hereditarily indecomposable continua. Proc. Amer. Math. Soc. 75 (1968) 502-503. 3. n-aposyndetic continua and cutting theorems. Trans. Amer. Math. Soc. 140 (1969) 127-135. 4. Stronger forms of aposyndetic continua. Proc. Topology Con- ference (Arizona State Univ., Tempe, Ariz., 1967) 170-173. Ari- zona State Univ., Tempe, Ariz., (1968)(with Jones, F. Burton)

Jones, F. Burton 01 • 1. One-to-one continuous images of a line. Fund. Math. 67 (1970) no. 3, 285-292. ACADEMIC DESCENDANTS AND PUBLICATIONS 309

2. Gordon T. Whyburn 1904-1969. Bull. Amer. Math. Soc. 77 (1971) 57-72.(with Floyd, E. E.)

McAuley, L. F. 02 • 1. Some fundamental theorems and problems related to monotone mappings. Proc. of the First Conf. on Monotone Mappings and Open Mappings. (editor) (SUNY at Binghamton, Binghamton, N. Y., 1970) State Univ. of New York at Binghamton, Bingham- ton, N.Y., 1971. 2. Spaces of certain non-alternating mappings. Duke Math. J. 38 (1971) 43-56. 3. Concerning open selections: Set-Valued Mappings, Selections, and Topological Properties of 2X .(Proc. Conf. SUNY, Buﬀalo, N.Y., 1969) 49-53. Lecture Notes in Mathematics Vol. 171. Springer, Berlin, 1970.

Ungar, Gerald S. 03 • 1. Spaces of homeomorphisms. Port. Math. 27 (1968) 67-73.

Wilson, D. C. 03 • 1. Complete regular mappings and dimension. Bull. Amer. Math. Soc. 76 (1970) 1057-1061. 2. Monotone mappings of manifolds onto cells. Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Bing- hamton, Binghamton, N. Y., 1970) State Univ. of New York at Binghamton, Binghamton, N. Y., 1971. 37-54.

Christoph, F. T. 03 • 1. Ideal extensions of topological semigroups. Canadian J. Math. 22 (1970) 1168-1175. 2. Embedding topological semigroups in topological groups. Semi- group Forum 1 (1970) no. 3, 224-231. 3. Free topological semigroups and embedding topological semigroups in topological groups. Paciﬁc J. Math. 34 (19703 343- 353.

Haver William E. 03 • 310 CHAPTER 9

1. A characterization theorem for cellular maps. Bull. Amer. Math. Soc. 76 (1970) 1277-1280.

Woodruﬀ, Edythe P. 03 • 1. Concerning the condition that a disk in E3/G be the image of a disk in E3. Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Binghamton, N.Y., 1970) State Univ. of New York at Binghamton, Binghamton, N. Y., 1971.

Hagopian, Charles L. 03 • 1. On generalized forms of aposyndeses. Paciﬁc J. Math. 34 (1970) 97-108. 2. A ﬁxed point theorem for plane continua. Bull. Amer. Math. Soc. 77 (1971) 351-354. 3. Errata to: ‘Concerning arcwise connectedness and the evidence of simple closed curves in plane continua.’ Trans. Amer. Math. Soc. 157 (1971) 507-509. 4. A class of arcwise connected continua. Proc. Amer. Math. Soc. 30 (1971) 164-168.

Heath, Robert W. 02 • 1. An easier proof that a certain countable space is not stratiﬁable. Proc. Wash. State Univ. Conf. on General Topology, Pullman, Wash., 1970) 56-59. Pi Mu Epsilon, Department of Mathemat- ics, Washington State University, Pullman, Washington, 1970.

Roy, Prabir 02 • 1. Separability of metric spaces. Trans. Amer. Math. Soc. 149 (1970) 19-43.

Thomas, E. S. 02 • 1. Flows on nonorientable 2-manifolds. J. Diﬀ. Equa. 7 (1970) 448-453. 2. The depth of the center 2-manifolds. Global Analysis (Proc. Sympos. Pure Math. Vol. XIV, Berkeley, Calif., 1968) 253- 264. American Mathematical Society, Providence, Rhode Island, 1970.(with Schwartz, A. J.) ACADEMIC DESCENDANTS AND PUBLICATIONS 311

Vought, E. J. 02 • 1. Concerning continua not separated by any nonaposyndetic subcontinuum. Paciﬁc J. Math. 31 (1969) 257-262.

Rogers, James T., Jr. 02 • 1. The pseudo-circle is not homogeneous. Trans. Amer. Math. Soc. 148 (1970) 417-428. 2. Mapping the pseudo-arc onto circle-like, self-entwined continua. Mich. Math. J. 17 (1970) 91-96. 3. Pseudo-circles and universal circularly chainable continua. Illi- nois J. Math. 14 (1970) 222-237. 4. Inverse limit spaces deﬁned by only ﬁnitely many distinct bonding maps. Fund. Math. 68 (1970) 117-120.(with Jolly, R. F.) 5. Embedding the hyperspaces of circle-like plane continua. Proc. Amer. Math. Soc. 29 (1971) 165-168. 6. Homeomorphism groups of weak solenoidal spaces. Proc. Amer. Math. Soc. 28 (1971) 242-246.(with Tollefson, J. L.) 7. Homogeneous inverse limit spaces with non-regular covering maps as bonding maps. Proc. Amer. Math. Soc. 29 (1971) 417-420. (with Tollefson, J. L.)

Shirley, E. D. 02 • 1. Homomorphisms on connected topological lattices. Duke Math. J. 38 (1971) 483-490.(with Stralka, A. R.)

Rogers, Leland 02 • 1. The computation of fundamental periodic subadditive functions. Math. Algorithms 3 (1968) 2-22.(with Jolly, R. F. Detmer, Richard C.)

Gordh, G. R., Jr. 02 • 1. Monotone decompositions of irreducible Hausdorﬀ continua. Pa- ciﬁc J. Math. 36 (1971) 647-658. 312 CHAPTER 9 R. H. Sorgenfrey and his Mathematical Descendandts

01 Sorgenfrey, R. H. 02 Whittaker, James, U. C. L. A., 1958 03 Su, Li Pi , University of British Columbia, 1966 03 Chew, Kim Peu, University of British Columbia, 1969 03 Lee, Jong Pil, University of Alberta, 1970 03 Choo, Eng Ung, University of British Columbia, 1971 02 Berri, Manua P., U. C. L. A., 1961 03 Scarborough, Charles T., Tulane University, 1964 03 Strecker, George E., Tulane University, 1966 03 Stephenson, Robert M., Jr., Tulane University, 1967 03 Schnare, Paul S., Tulane University, 1967 02 Franklin, Stanley P., U. C. L. A., 1963 03 Mishra, Arvind Kumar, Indian Institute of Technology, 1970 03 Nyikos, Peter, Carnegie-Mellon University, 1971 02 Sabello, Ralph, U. C. L. A., 1969 ACADEMIC DESCENDANTS AND PUBLICATIONS 313 Publications of R. H. Sorgenfrey and his Mathematical Descendants

Sorgenfrey, R. H. 01 • 1. Concerning triodic continua. Amer. J. Math. 66 (1944) 439-460. 2. Some theorem on co-terminal arcs. Bull. Amer. Math. Soc. 50 (1944) 257-259. 3. Concerning continua irreducible about N points. Amer. J. Math. 68 (1946) 667-671. 4. On the topological product of paracompact spaces. Bull. Amer. Math. Soc. 53 (1947) 631-632. 5. Dimension lowering mappings of convex sets. Proc. Amer. Math. Soc. 5 (1954) 179-181. 6. Minimal regular spaces. Proc. Amer. Math. Soc. 14 (1963) 454-458.(with Berri, M. P.) 7. Closed and image-closed relations. Paciﬁc J. Math. 19 (1966) 433-439.(with Franklin S P.)

Whittaker, James V. 02 • 1. On the structure of half-groups. Canadian J. Math. 11 (1959) 651659. 2. Normal subgroups of some homeomorphism groups. Paciﬁc J. Math. 10 (1960) 1469-1478. 3. Coincidence sets and transformations of function spaces. Trans. Amer. Math. Soc. 101 (1961) 457-476. 4. On isomorphic groups and homeomorphic spaces. Ann. of Math. (2) 78 (1963) 74-91. 5. Some normal subgroups of homeomorphisms. Trans. Amer. Math. Soc. 123 (1966) 88-98. 6. A mountain-climbing problem. Canadian J. Math. 18 (1966) 873-880. 7. Multiply transitive groups of transformations. Paciﬁc J. Math. 23 (1967) 180-207.

Su, Li Pi 03 • 1. Algebraic properties of certain rings of continuous functions. Pa- ciﬁc J. Math. 27 (1968) 175-191. 314 CHAPTER 9

Chew, K. P. 03 • 1. Distance group. Bull. Math. Soc. Nanyang Univ. (1964) 75- 100.

Lee, Jong Pil 03 • 1. Semi-topological groups. Amer. Math. Monthly 72 (1965) 996- 998.(with Bohn, Elwood)

Choo, Eng Ung 03 • 1. Some results on coherent graphs. J. Nanyang Univ. 1 (1967) 289-308.(with Teh, H. H.)

Berri, Manual P. 03 • 1. Minimal topological spaces. Trans. Amer. Math. Soc. 108 (1963) 97-105. 2. Categories of certain minimal topological spaces. J. Austral. Math. Soc. 4 (1964) 78-82. 3. The complement of a topology for some topological groups. Fund. Math. 58 (1966) 159-162. 4. Minimal regular spaces. Proc. Amer. Math. Soc. 4 (1964) 7882.(with Sorgenfrey, R. H.)

Scarborough, Charles T. 03 • 1. Minimal Urysohn spaces. Paciﬁc J. Math. 27 (1968) 611-617. 2. Minimal topologies. Colloq. Math. 19 (1968) 215-219. (with Stephenson, R. M.) 3. Products of nearly compact spaces. Trans. Amer. Math. Soc. 124 (1966) 131-147.(with Stone, A. H.)

Strecker, G. E. 03 • 1. The compactness operator in general topology. General topology and its relations to modern analysis and algebra, II (Proc. Second Prague Topological Sympos., 1966) 161-163. Acadmia, Prague, 1967.(with DeGroot, J.; Wattel, E.) 2. H-closed spaces and reﬂective sub-categories. Math. Ann. 177 (1968) 302-309.(with Herrlich, H.) ACADEMIC DESCENDANTS AND PUBLICATIONS 315

3. Cotopology and minimal Hausdorﬀ spaces. Proc. Amer. Math. Soc. 21 (1969) 569-574.(with Higliono, G.) 4. A coherent embedding of an arbitrary topological space in a semiregular space. Math. Centrum. Amsterdam Afd. Zuivere Wisk. (1966) ZW-006, 11.(with Wattel, E.) 5. On semi-regular and minimal Hausdorﬀ embeddings. Neder. Akad. Wetensch. Proc. Ser. A 70 = Idag. Math. 29 (1967) 234-237. 6. J. Strengthening Alexander’s subbase theorem. Duke Math. J. 36 (1968) 671-676.(with Wattel, E.; Herrlich, H; de Groot, J.)

Stephenson, R. M., Jr. 03 • 1. Spaces for which the Stone-Weierstrass theorem holds. Trans. Amer. Math. Soc. 133 (1968) 537-546. 2. Pseudocompact spaces. Trans. Amer. Math. Soc. 134 (1968) 437448. 3. Two minimal first countable Hausdorff spaces. Math Zeit. 108 (1969) 171-i72. 4. Minimal first countable topologies. Trans. Amer. Math. Soc. 138 (1969) 115-127. 5. A countable minimal Urysohn space is compact. Proc. Amer. Math. Soc. 22 (1969) 625-626. 6. Noncut points and modified compactness conditions. Proc. Amer. Math. Soc. 23 (1969) 266-272. 7. Product spaces for which the Stone-Weierstrass theorem holds. Proc. Amer. Math. Soc. 21 (1969) 284-288. 8. Minimal topologies. Colloq. Math. 19 (1968) 215-219. (with Scarborough, C. T.)

Jensen, R. A. 02 • 1. Surfaces of vertical order 3 are tame. Bull. Amer. Math. Soc. 76 (1970) 151-154.(with Loveland, L. D.)

Schnare, Paul S. 03 • 1. Multiple complementation in the lattice of topologies. Fund. Math. 62 (1968) 53-59. 316 CHAPTER 9

2. The maximal to (Respectively T1) subspace lemma is equivalent to the axiom of choice. Amer. Math. Monthly 75 (1968) 761. 3. Inﬁnite complementation to the lattice of topologies. Fund. Math. 64 (1969) 249-255.

Franklin, S. P. 02 • 1. Quotient topologies from power topologies. Arch. Math. 15 (1964) 341-342. 2. Spaces in which sequences suffice. Fund. Math. 57 (1965) 107115. 3. Compactness and semi-continuity. Israel J. Math. 3 (1965) 13- 14. 4. On unique sequential limits. Nieuw Arch. Wisk. (3) 14 (1966) 12-14. 5. Open and image-open relations. Colloq. Math. 12 (1964) 209- 211. 6. Spaces in which sequences suffice. II. Fund. Math. 61 (1967) 51-56. An isomorphism theorem. Proc. Indian Acad. Sci. Sect. A 67 (1968) 219-221. 7. Natural covers. Compositio Math. 21 (1969) 253-291. 8. On two questions of Moore and Mrowka. Proc. Amer. Math. Soc. 21 (1969) 597-599 9. Ordinal invariants for topological spaces. Mich. Math. J. 15 (1968) 313-320, Addendum, ibid. 15 (1968) 506.(with Archangel- skii, A. V.) 10. Spaces of continuous relations. Math. Ann. 169 (1967) 289293.(with Day, Jane M.) 11. Straddles on semigroups. Math. Mag. 34 (1960/61) 269-270. (with Lindsay, John W.) 12. 0-sequences and 0-nets. Amer. Math. Monthly 72 (1965) 506510.(with Robertson, L. C.) 13. Closed and image-closed relations. Pacific J. Math. 19 (1966) 433-439.(with Sorgenfrey, R. H.) 14. The least element map. Colloq. Math. 15 (1966) 217-221. (with Wallace, A. D.)

Su, L. P. 03 • ACADEMIC DESCENDANTS AND PUBLICATIONS 317

1. A homology theorem for rings of functions. Amer. Math. Monthly 76 (1969) 804-806.

Chew, K. P. 03 • 1. A characterization of N-compact spaces. Proc. Amer. Math. Soc. 26 (1970) 679-682.

Berri, M. P. 02 • 1. A survey of minimal topological spaces. General Topology and its Relations to Modern Analysis and Algebra. III (Proc. Conf. Kanpur, 1968) 93-114. Academie, Prague, 1971.(with Porter, J. R.; Stephenson, R. M., Jr.)

Strecker, G. E. 03 • 1. Compactness as an operator. Composito Math. 21 (1969) 349- 375. (with DeGroot, J.; Herrlich, H.; Wattel, E.) 2. Coreﬂective subcategories. Trans. Amer. Math. Soc. 157 (1971) 205-226.(with Herrlich, H.)

Franklin S. P. 02 • 1. On open extensions of maps. Canadian J. Math. 22 (1970) 691-696. (with Kohli, J. K.) 2. On the topological characterization of the real line. J. London Math. Soc. (2) 2 (1970) 589-591.(with Krishnarao, G. V.) 3. A homogeneous Hausdorﬀ Eo space which is not El; a new metrization proof. Proc. of the Kanpur Topological Conf., 1968. Scientiﬁc editors: J. Novak, M. Venkataraman, G. T. Whyburn. Editors: S. P. Franklin, A. Frolik, V. Koutnik. Academia [Pub- lishing House of the Czech. Acad. of Sci], Prague; Academic Press, New York-London, 1971. 332 pp.

Publications of R. L. Swain

Swain R. L. 01 • 1. Approximate isometries in bounded spaces. Proc. Amer. Math. Soc. 2 (1951) 727-729. 318 CHAPTER 9

2. Bounded models of the Euclidean plane. I. Condensed graphs. Amer. Math. Monthly 61 (1954) 21-26.

Publications of Harlan C. Miller

Miller, Harlan C. 01 • 1. A theorem concerning closed and compact point sets which lie in connected domains. Bull. Amer. Math. Soc. 46 (1940) 848. 2. On unicoherent continua. Trans. Amer. Math. Soc. 69 (1950) 179194. ACADEMIC DESCENDANTS AND PUBLICATIONS 319 Gail S. Young and his Mathematical Descendandts

01 Young, Gail S. 02 Alford, William Robert, Tulane University, 1963 03 Mullin, J. B., Georgia University, 1970 02 Haque, Mohammed Rashidul, Tulane University, 1964 02 Marx, Morris Leon, Tulane University, 1964 03 Wells, Carroll Glenn, Vanderbilt University, 1969 03 Kilgore, Mary Spruill, Vanderbilt University, 1969 02 Mathews, Harry Thomas, Tulane University, 1969 03 Rodriguez, Rene, University of Tennessee, 1968 03 Cowan, Richard, Universityof Tennessee, 1969 03 Chadick, Stanley, University of Tennessee, 1969 02 Schneider, Walter Jan, Tulane University, 1964 02 Boyce, William Martin, Tulane University, 1967 02 Cannon, Raymond Joseph, Jr., Tulane University, 1968 320 CHAPTER 9 Publications of Gail S. Young and his Mathematical Descendants

Young, Gail S. 01 • 1. A generalization of Moore’s theorem of simple triods. Bull. Amer. Math. Soc. 50 (1944) 714. 2. On continua whose links are non-intersecting. Bull. Amer. Math. Soc. 50 (1944) 920-925. 3. Spaces in which every arc has two sides. Ann. of Math. (2) 46 (1945) 182-193. 4. The introduction of local connectivity by change of topology. Amer. J. Math. 68 (1946) 479-494. 5. Spaces congruent with bounded subsets of the line. Bull. Amer. Math. Soc. 52 (1946) 915-917. 6. On compact ﬁberings of the plane. Bull. Amer. Math. Soc. 53 295-298 (1947) (Clayton) 8-595. 7. A characterization of 2-manifolds. Duke Math. J. 14 (1947) 979990 . 8. On l-regular convergence of sequences of 2-manifolds. Amer. J. Math. 71 (1949) 339-343. 9. On continuous curves irreducible about compact sets. Bull. Amer. Math. Soc. 55 (1949) 439-441. 10. On the factors and ﬁberings of manifolds. Proc. Amer. Math. Soc. 1 (1950) 215-223. 11. A generalization of the Rutt-Roberts theorem. Proc. Amer. Math. Soc. 2 (1951) 586-588. 12. A footnote to ”Statistical decision functions.” Mich. Math. J. (1952) 186-188 (1953). 13. The linear functional equation. Amer. Math. Monthly 65 (1958) 37-38. 14. A theorem on dimension. Proc. Amer. Math. Soc. 3 (1952) 159-161 (Katetov) 13-764.(with Curtis, Morton L.) 15. Extensions of Liouville’s theorem to n dimensions. Math. Scand. 6 (1958) 289-292. 16. A generalization of Bagemihl’s theorem of ambiguous points. Mich. Math. J. 5 (1958) 223-227. ACADEMIC DESCENDANTS AND PUBLICATIONS 321

17. Fixed-point theorems arcwise connected continua Proc. Amer. Math. Soc. 11 (1960) 880-884. 18. Types of ambiguous behavior of analytic functions. Mich. Math. J. 8 (1961) 193-200. 19. Representations of Banach spaces. Proc. Amer. Math. Soc. 13 (1962) 667-668. 20. Types of ambiguous behavior of analytic functions. II. Mich. Math. J. 10 (1963) 147-149. 21. The inversion of Peano continua by analytic functions. Fund. Math. 56 (1964-65) 301-311. 22. A condition for the absolute homotopy extension property. Amer. Math. Monthly 71 (1964) 896-897. 23. Topology. Addison-Wesley Publishing Co., Reading, Mass. and London (1961) ix+374.(with Hocking, J. G.) 24. Product spaces in n-manifolds. Proc. Amer. Math. Soc. 10 (1959) 307-308.(with Jones, F. B.) 25. Conformal mapping and Peano curves. Mich. Math. J. 1 (1952) 69-72 (Rogosinski) 14-262.(with Piranian, George; Titus, C. J.) 26. A Jacobian condition for interiority. Mich. Math. J. 1 (1952) 89-94 (G. T. Whyburn) 14-192.(with Titus, C. J.) 27. An extension theorem for a class of diﬀerential operators. Mich. Math. J. 6 (1959) 195-204.(with Titus, C. J.) 28. The extension of interioritys with some applications. Trans. Amer. Math. Soc. 103 (1962) 329-340.

Alford, W. R. 02 • 1. Some ”nice” wild 2-spheres in E3. Topology of 3-manifolds and related topics. (Proc. The Univ. of Georgia Institute, 1961) 29-33 Prentice-Hall, Englewood Cliﬀs, N. J., 1962. 2. Uncountably many diﬀerent involutions of S3. Proc. Amer. Math. Soc. 17 (1966) 186-196. 3. Complements of minimal spanning surfaces of knots are not unique. Ann. of Math. (2) 91 (1970) 419-424. 4. Some almost polyhedral wild arcs. Duke Math. J. 30 (1963) 33-38.(with Ball, B. J.) 5. A note on 0-dimensional decompositions of E3. Amer. Math. Monthly 75 (1968) 377-378.(with Sher, R. B.) 322 CHAPTER 9

6. Deﬁning sequences for compact O-dimensional decompositions of En . Bull. Acad. Polon. Sci. Ser. Scie. Math. Astronom. Phys. 17 (1969) 209-212.(with Sher, R. B.)

Haque, M. R. 02 • 1. Cech homology and cohomology groups of compact, lcn spaces. I. J. Natur. Sci. and Math. 7 (1967) 59-70. 2. On a theorem of G. Ricci Curbastro and the Gaussian curvatures of the minimal surfaces in R4. J. Natur. Sci. and Math. 7 (1967) 245-251. 3. A duality theorem of Pontrjagin type. J. Natur. Sci. and Math. 7 (1967) 51-57.

Marx, Morris L. 02 • 1. Normal curves arising from light open mappings of the annulus. Trans. Amer. Math. Soc. 120 (1965) 46-56. 2. The branch point structure of extension of interior boundaries. Trans. Amer. Math. Soc. 131 (1968) 79-98. 3. Whyburn’s conjecture for C2 maps. Proc. Amer. Math. Soc. 19 (1968) 660-661. 4. Light open mappings on a torus with a disc removed. Mich. Math. J. 15 (1968) 449-456. 5. The Gauss realizability problem. Proc. Amer. Math. Soc. 22 (1969) 610-613. 6. Interior and polynomial extensions of immersed circles.(with Ver- hey, Roger F.) 7. Linear recursive sequences. SIAM Rev. 10 (1968) 342-353. (with Fillmore, Jay P.)

Mathews, H. T. 02 • 1. A note on Bagemihl’s ambiguous point theorem. Math Zeit. 9 tl965) 138-139. 2. Cluster sets at isolated and nonisolated singularities. Proc. Nat. Acad. Sci. 53 (1965) 1264-1266. 3. Left and right boundary cluster sets in n-space. Duke Math. J. 33 (1966) 667-672. ACADEMIC DESCENDANTS AND PUBLICATIONS 323

4. The n-arc property for functions meromorphic in the disk. Math Zeit. 93 (1966) 164-170.

Schneider W. J. 02 • 1. An entire transcendental function whose inverse takes sets of finite measure into sets of finite measure. Bull. Amer. Math. Soc. 72 (1966) 841-842. 2. Some extensions of the Nevanlinna 2-constant theorem and the Hadamard 3-circle thecrem. J. Math. Anal. Appl. 17 (1967) 280291. 3. A problem in 2-dimensional heat flow. Math. Mag. 40 (1967) 144145. 4. An elementary proof of a theorem of MacLane. Monatsh. Math. 72 (1968) 144-146. 5. On the growth of entire functions along half rags. Entire functions and related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Amer. Math. Soc. Providence, R. I., 1968, 377-385. 6. A note on L. D. Faddeev’s three-particle theory. Helv. Phys. Acta 40 (1967) 745-748.(with Albeverio, S,; Hunziker, W. and Schrader, R.) 7. A short proof of a lemma of G. R. MacLane. Proc. Amer. Math. Soc. 20 (1969) 604-605.(with Barth, K. F.) 8. On a problem of Bagemihl and Erdos concerning the distribu- tions of zeros of an annular function. J. Reine Angew. Math. 234 (1969) 179-183. Exponentiation of functions in MacLane’s class A. J. Reine Angew. Math. 236 (1969) 120-130. 9. On the impossibillty of extending the Riesz uniqueness theorem to functions of slow growth. Ann. Acad. Sci. Fenn. Ser. A I No. 432 (1968) 9 pages. 10. On the question of Seidel concerning holomorphic functions bounded on a spiral. Canadian J. Math. 21 (1969) 1255-1262. 11. On a problem of Collingwood concerning meromorphic functions with no asymptotic values. J. London Math. Soc. (2) 1 (1969) 553560. 12. An asymptotic analog of the F. and M. Riesz radial uniqueness theorem. Proc. Amer. Math. Soc. 22 (1969) 53-54. 324 CHAPTER 9

13. Polynomial images of circles and lines. Math. Mag. 40 (1967) 1-4.(with Cargo, G. T.) Boyce, William M. 02 • 1. Commuting functions with no common fixed point. Trans. Amer. Math. Soc. 137 (1969) 77-92. 2. Generation of a class of permutations associated with commuting functions. Math. Algorithms 2 (1967) 19-26; Addendum, ibid. (1968) 25-26. Cannon, R. J., Jr. 02 • 1. Quasiconformal structures and the metrization of 2-manifolds. Trans. Amer. Math. Soc. 135 (1969) 95-103. 2. The determination of unknown parameters in analytic systems of ordinary differential equations. SIAM J. Appl. Math. 15 (1967) 799-809.(with Filmer, D. L.) 3. Remarks on a Stefan problem. J. Math. Mech. 17 (1967) 433- 441. (with Hill, C. Denson) 4. Continuous dependence of bounded solutions of a linear parabolic partial differential equation upon interior Cauchy date. Duke Math. J. 35 (1968) 217-230. Young, G. S. 01 • 1. Locally sense-preserving mappings with some applications to partial differential equations. Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Bingham- ton, N. Y., 1970) State University of New York at Binghamton, Binghamton, N. Y., 1971. Boyce, William M. 02 • 1. Γ-compact maps on an interval and fixed points. Trans. Amer. Math. Soc. 160 (1971) 87-102. Schneider, Walter J. 02 • 1. Entire functions mapping countable dense subsets of the real onto each other monotonically. J. London Math. Soc. (2) 2 (1970) 620-626.(with Barth K. F.) 2. On the shape of the level loci of harmonic measure. J. Analyse Math. 23 (1970) 441-460.(with Walsh, J. L.) ACADEMIC DESCENDANTS AND PUBLICATIONS 325 R. H. Bing and his Mathematical Descendandts

01 Bing, R. H. 02 Cohen, Herman, University of Wisconsin, 1942 02 Thomas, Garth, University of Wisconsin, 1952 02 Sanderson, Donald, University of Wisconsin, 1953 03 Jones, R. E. D., Iowa State University, 1962 03 Schmidt, D. L., Iowa State University, 1962 04 Steﬀenson, Arnold R., Univ. of Northern Colorado, 1968 04 Kieft, Raymond N., Univ. of Northern Colorado, 1969 04 Smith, Kenneth A., Univ. of Northern Colorado, 1969 04 McKinley, William S., Univ. of Northern Col., 1969 04 Bushnell, Donald D., Univ. of Nothern Col., 1969 04 Werremeyer, Frederic N., Univ. of Northern Col., 1971 03 Hildebrand, S. K., Iowa State University, 1962 04 Crossley, S. Gene, Texas Technological Univ., 1968 03 Sims, B. T., Iowa State University, 1962 03 Robinson, T. J., Iowa State University, 1963 03 Irudayanathan, A., Iowa State University, 1967 03 McCoy, R. A., Iowa State University, 1968 03 Curtis, D. W., Iowa State University, 1968 03 Fischer, D. R., Iowa State University, 1971 02 Goblirsch, Robert, University of Wisconsin, 1956 02 Lehner, Guydo, University of Wisconsin, 1958 03 Bean, Ralph, University of Maryland, 1962 04 Bailey, John Lay, University of Tennessee, 1968 04 Boyd, William S., University of Tennessee, 1969 03 Hyman, Daniel, University of Maryland, 1966 03 Reed, James, University of Maryland, 1969 02 Brown, Morton, University of 7isconsin, 1958 02 Kister, James, University of Wisconsin, 1959 03 Bennett, Albert, University of Michigan, 1966 03 Miller, Richard, University of Michigan, 1968 03 Edwards, Richard, University of Michigan, 1969 03 Strube, Richard, University of Michigan, 1970 03 Connelly, Robert, University of Michigan, 1970 02 Rosen, Ronald, University of Wisconsin, 1959 02 McMillan, Daniel, University of Wisconsin, 1960 03 Slack, Stephen P., University of Wisconsin, 1968 326 CHAPTER 9

03 Jaco, William H., University of Wisconsin, 1968 04 Lyon, Herbert C., University of Michigan, 1970 04 Evans, Benny Dan, University of Michigan, 1971 03 Row, W. Harold, University of Wisconsin, 1969 03 Wright, Alden H., University of Wisconsin, 1969 02 Gillman, David, University of Wisconsin, 1962 03 Ferris, Ian, U. C. L. A., 1968 02 Hempel, John, University of Wisconsin, 1962 03 Belfi, Victor, Rice University, 1969 03 Heil, Wolfgang H., Rice University, 1970 03 Roeling, Lloyd G., Rice University, 1970 02 Casler, Burtis Griffin, University of Wisconsin, 1962 02 Glaser, Leslie, University of Wisconsin, 1964 02 Hosay, Norman, University of Wisconsin, 1964 02 Price, Thomas, University of Wisconsin, 1964 03 Nicholson, Victor A., University of Iowa, 1968 03 Dieffenbach, Robert M., University of Iowa, 1971 02 Henderson, David, University of Wisconsin, 1964 03 Geoghagen, Ross, Cornell University, 1970 03 Cutler, William, Cornell University 1970 02 Dancis, Jerome, University of Wisconsin, 1966 03 Lusk, Ewing L., University of Maryland, 1970 02 Cobb, John, University of Wisconsin, 1966 02 McAllister, Byron, University of Wisconsin, 1966 02 Craggs, Robert, University of Wisconsin, 1967 02 Jones, Stephen, University of Wisconsin, 1967 02 Yohe, Michael, University of Wisconsin, 1967 02 Wright, Perrin T., University of Wisconsin, 1967 02 Daverman, Robert, University of Wisconsin, 1967 03 Bass, Charles D., University of Tennessee, 1971 02 Jensen, Richard, University of Wisconsin, 1968 02 Cash, Burt, University of Wisconsin, 1969 02 01 inick, M., University of Wisconsin, 1970 02 Gerlach, Jacob, University of Wisconsin, 1970 02 Webster, Dallas E., University of Wisconsin, 1970 ACADEMIC DESCENDANTS AND PUBLICATIONS 327 Publications of R. H. Bing and his Mathematical Descendants

Bing, R. H. 01 • 1. Collections ﬁlling up a simple plane web. Bull. Amer. Math. Soc. 51 (1945) 674-679. 2. Generalizations of two theorems of Janiszewski. Bull. Amer. Math. Soc. 51 (1945) 945-960. 3. Generalizations of two theorems of Janiszewski. II. Bull. Amer. Math. Soc. 52 (1946) 478-480. 4. Converse linearity conditions. Amer. J. Math. 68 (1946) 309318. 5. Concerning simple plane webs. Trans. Amer. Math. Soc. 60 (1946) 133-148. 6. Sets cutting the plane. Annals. of Math 47 (1946) 476-479. 7. The Kline sphere characterization problems. Bull. Amer. Math. Soc. 52 (1946) 644-653. 8. Extending a metric. Duke Math. J. 14 (1947) 511-519. 9. Skew sets. Amer. J. Math. 69 (1947) 493-498. 10. Solution of a problem of R. L. Wiider. Amer. J. Math. 70 (1948) 95-98. 11. Some characterizations of arcs and simple closed curves. Amer. J. Math. 70 (1948 497-506. 12. A homogeneous indecomposable plane continuum. Duke Math. J. 15 (1948) 729-742. 13. A convex metric for a locally connected continuum. Bull. Amer. Math. Soc. 55 (1949) 812-819. 14. Partitioning a set. Bull. Amer. Math. Soc. 55 (1949) 1101- 1110. 15. Complementary domains of continuous curves. Fund. Math. 36 (1949) 303-318. 16. Coverings with connected intersections. Trans. Amer. Math. Soc. 69 (1950) 387-391.(with Floyd, E. E.) 17. A characterization of 3-space by partitioning. Trans. Amer. Math. Soc. 70 (1951) 15-27. 18. Metrization of topological spaces. Canadian J. Math. 3 (1951) 175-186. 328 CHAPTER 9

19. An equilateral distance. Amer. Math. Monthly 58 (1951) 380- 383. 20. Concerning hereditarily indecomposable continua. Paciﬁc J. Math. (1951) 43-51. 21. Higher dimensional hereditarily indecomposable continua. Trans. Amer. Math. Soc. 71 (1951) 267-273. 22. Snake-like continua. Duke Math. J. 18 (1951) 653-663. 23. Partitioning continuous curves. Bull. Amer. Math. Soc. 58 (1952) 536-556. 24. The sum of two horned spheres. Annals. of Math. 56 (1952) 354362. 25. A convex metric with unique segments. Proc. Amer. Math. Soc. 4 (1953) 167-174. 26. A countable connected Hausdorﬀ space. Proc. Amer. Math. Soc. 4 (1953) 474. 27. Examples and counterexamples. Pi Mu Epsilon J. 1 (1953) 311- 317. 28. Locally tame sets are tame. Annals. of Math. 56 (1954) 145-158. 29. Review of General Topology by W. Sierpinski. Bull. Amer. Math. Soc. 59 (1953) 410. 30. Partially continuous decompositions. Proc. Amer. Math. Soc. 6 (1955) 124-133. 31. Some monotone decompositions of a cube. Annals. of Math. 61 (1955) 279-288. 32. What topology is here to stay? Summary of lectures and seminars. Summer Inst. on Set Theoretic Topology, Madison, Wis., (1955) 25-27. 33. Decomposition of E3 into points and tame arcs. Summary of lectures and seminars. Summer Inst. on Set Theoretic Topology, Madison, Wis., (1955) 40-47. 34. Approximating surfaces with polyhedral ones. Summary of lectures and seminars. Summer Inst. on Set Theoretic Topology, Madison, Wis., (1955) 47-52. 35. The pseudo-arc. Summary of lectures and seminars. Summer Inst. on Set Theoretic Topology, Madison, Wis., (1955) 70-73. 36. Point set topology. Insights into Modern Mathematics; Chapter X. Yearbook of the National Council of Teachers of Mathematics (1957). ACADEMIC DESCENDANTS AND PUBLICATIONS 329

37. Upper semicontinuous decompositions of E3. Annals. of Math. 65 (1957) 363-374. 38. A simple closed curve that pierces no disk. J. de Math. Pures et Appl. 35 (1956) 337-343. 39. A decomposition of E3 into points and tame arcs that the decomposition space is topologically different from E3. Annals. of Math. 65 (1957) 484-500. 40. Approximating surfaces with polyhedral ones. Annals. of Math. 65 (1957) 456-483. 41. Necessary and sufficient conditions that a 3-manifold be S3. An- nals of Math. 68 (1959) 17-37. 42. An alternate proof that 3-manifolds can be triangulated. Annals of Math. 69 (1959) 37-65. 43. Another homogeneous plane continuum. Trans. Amer. Math. Soc. 90 (1959) 171-192.(with Jones, F. B.) 44. The Cartesian product of a certain non-manifold and a line is E4. Bull. Amer. Math. Soc. 64 (1958) 82-84. 45. Each homogeneous nondegenerate chainable continuum is a pseudo- arc. Proc. Amer. Math. Soc. 10 (1959) 345-346. 46. Conditions under which a surface in E3 is tame. Fund. Math. 47 (1959) 105-139. 47. The Cartesian product of a certain non-manifold and a line is E4. Annals. of Math. 70 (1959) 399-412. 48. A simple closed curve is the only homogeneous bounded plane continuum that contains an arc. Canadian J. Math. 12 (1960) 209-230. 49. Elementary point set topology. Number 8 of the Herbert Ellsworth Slaught Memorial Papers. Amer. Math. Monthly 67 (1960) No. 7, 58. 50. Set Theory. McGraw Hill Encyclopedia of Science and Technol- ogy, McGraw-Hill Book Co. (1960) 205-206. 51. A set is a 3-cell if its cartesian product with an arc is a 4-cell. Proc. Amer. Math. Soc. 12 (1961) 13-19. 52. A wild surface each of whose arcs is tame. Duke Math. J. 28 (1961) 1-16. 53. Tame Cantor sets in E3. Pacific J. Math. 11 (1961) 435-446. 330 CHAPTER 9

54. A surface is tame if its complement is l-ULC. Trans. Amer. Math. Soc. 101 (1961) 294-305. 55. Embedding circle like continua in the plane. Canadian J. Math. 14 (1962) 113-128. 56. Decompositions of E3. Topology of 3-Manifolds and Related Topics. Prentice Hall (1962) 5-21. 57. Applications of the side approximation theorem for surfaces. Proc. Symp. on General Topology and its Relation to Modern Analysis and Algebra, Prague (1961) 91-95. 58. Approximating surfaces from the side. Annals. of Math. 77 (1963) No. 1, 145-192. 59. Point-like decompositions of E3. Fund. Math. 50 (1962) 431- 453. 60. Necessary and suﬃcient conditions that a 3-manifolds be S3. Annals of Math. 77 (1963) 210. 61. Each disk in each 3-manifold is pierced by a tame arc. Amer. J. Math. 84 (1962) No. 4, 591-599. 62. Embedding surfaces in 3-manifolds. Proc. International Congress of Math. Stockholm, 1963. 63. Pushing a 2-sphere into its complement. Mich. Math. J. 11 (1964) 33-45. 64. Retractions onto spheres. Amer. Math. Monthly 71 (1964) 481484. 65. Spheres in E3. Amer. Math. Monthly 71 (1964) 353-364. 66. Some aspects of the topology of 3-manifolds related to the Poincare conjectures. Lectures in Modern Mathematics(John Wiley & Sons) 2 (1964) Chapter, 93-128. 67. Inequivalent families of periodic homeomorphisms. Annals. of Math. 80 (1964) 78-93. 68. The simple connectivity of the sum of two discs. Paciﬁc J. Math. 14 (1964) 79-93. 69. An arc is tame in 3-space if and only if it is strongly cellular. Fund. Math. 55 (1964) 175-180.(with Kirkor, A.) 70. Some remarks concerning topologically homogeneous spaces. An- nals of Math. 81 (1965) 100-111. 71. A translation of the normal Moore space conjecture. Proc. Amer. Math. Soc. 16 (1965) No. 4, 612-619. ACADEMIC DESCENDANTS AND PUBLICATIONS 331

72. Improving the side approximation theorem. Trans. Amer. Math. Soc. 116 (1965) Issue 4, 511-525. 73. Computing the fundamental group of the complements of curves. Tech. Report 2, Washington State Univ. (1965) 20 pages. 74. Challenging conjectures. Amer. Math. Monthly 74 (1967) Part II, 56-64. 75. Improving the intersections of lines and surfaces. Mich. Math. J. 14 (1967) 155-159. 76. A hereditarily infinite dimensional space. General Topology and Its Relations to Modern Analysis and Algebra II. Second Prague Topological Symposium (1966) 56-62. 77. Geometry, Mathematics for High School. School Mathematics Study Group (1959), joint author. 78. A finite number of reflection operations render a nonconvex poly- gon connex. Mat. Prov., No. 8 (1961) (written in Russian by Boltyanski.(with Kazarinoff, N. D.) 79. The place of topology in a teacher training program. Symposium on Teacher Education in Mathematics, Madison, Wis., (1962) 4 pages. 80. A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines. Bull. Amer. Math. Soc. 74 (1968) No. 5, 771-792. 81. The elusive fixed point property. Amer. Math. Monthly 76 (1969) No. 2, 119-132. 82. Mapping a 2-sphere onto a homotopy 2-sphere. Wisconsin Topol- ogy Seminar Notes, 1965, Princeton Univ. Press (edited by R. H. Bing and R. J. Bean), (1966) 89-99. 83. Radial engulfing. Conference on the Topology of Manifolds. Prindle, Weber, and Schmidt (edited by John G. Hocking), (1968) 1-18. 84. Retractions onto ANR’s. Proc. Amer. Math. Soc., 21 (1969) No. 3, 618-620. 85. Point-set topology. The Mathematical Sciences Essays for COS- RIMS. M.I.T. Press (1969), edited by COSRIMS, 209-216. 86. Each disk in E3 contains a tame arc. Amer. J. Math. 84 (1962) 583-590. 87. Topology of 3-manifolds. Notes of the University of Wyoming Colloquium Series (distributed by A.M.S.), Laramie, Wyoming, August 24-28, 1970. Seventy-fifth summer meeting of the A.M.S., 30 pages. 332 CHAPTER 9

88. The elusive ﬁxed point property. Sugaku 21, No. 3 (1969) 203- 210. 89. The monotone mapping problem. Topology of 3-manifolds. Proc. of the Univ. of Georgia Topology of Manifolds Institute, 1969. Cantrell and Edwards, editors. Markham Pub. Co. 99-115. 90. A toroidal decomposition of E3. Fund. Math. 60 (1967) 81-87. (with Armentrout, Steve) 91. Conformal minimal varieties. Duke Math. J. 10 (1943) 637-740. (with Beckenbach, E. F.) 92. On generalized convex functions. Trans. Amer. MaLh. Soc. 58 (1945) 220-230.(with Beckenbach, E. F.) 93. A 3-dimensional absolute retract which does not contain any disc. Fund. Math. 54 (1964) 159-175.(with Borsuk, K.) 94. Imbedding and decompositions of E3 in E4. Proc. Amer. Math. Soc. 11 (1960) 149-155.(with Curtis, M. L.) 95. Every simple closed curve in E3 is unknotted in E4. Proc. Lon- don Math. Soc. 39 (1964) 86-94.(with Klee, Vic.) 96. Taming complexes in hyperplanes. Duke Math. J. 31 (1964) 491-512.(with Kister, J. M.) 97. Cubes with knotted holes. Trans. Amer. Math. Soc. 155 (1971) 653-681.(with Martin, J. M.) 98. Monotone images of E3. Proc. of Binghamton Conference on Monotone Mappings and Open Mappings, 1970.

Cohen, H. J. 02 • 1. Some results concerning homogeneous plane continua. Duke Math. J. 18 (1951) 467-474. 2. Sur un probleme de M. Dieudonne. C. R. Acad. Sci. Paris 234 (1952) 290-292. 3. Concerning real numbers whose powers have nonintegral diﬀer- ences. Proc. Amer. Math. Soc. 14 (1963) 626-627.(with Sup- nick, Fred) 4. On the powers of a real number reduced modulo one. Trans. Amer. Math. Soc. 94 (1960) 244-257.(with Supnick, Fred; Ke- ston, J. F.)

Thomas, Garth 02 • ACADEMIC DESCENDANTS AND PUBLICATIONS 333

1. Simultaneous partitioning of two sets. Trans. Amer. Math. Soc. 75 (1953) 69-79.

Sanderson, Don E. 02 • 1. Isotopy in 3-manifolds. I. Isotopic deformations of 2-cells and 3-cells. Proc. Amer. Math. Soc. 8 (1957) 912-922. 2. Isotopy in 3-manifolds. II. Fitting homeomorphisms by isotopy. Duke Math. J. 26 (1959) 387-396. 3. Isotopy in 3-manifolds. III. Connectivity of spaces of homeomorphisms. Proc. Amer. Math. Soc. 11 (1960) 171-176. 4. Isotopy in manifolds. Topology of 3-manifolds and related topics. (Proc. The Univ. of Georgia Instiute, 1961), 226-228. Prentice- Hall, Englewood Cliﬀs, N. J., 1962. 5. Relations among some basic properties of noncontinuous functions. Duke Math. J. 35 (1968) 407-414. 6. Connectivity functions and retracts. Fund. Math. 57 (1965) 237-245.(with Hildebrand, S. K.)

Jones R. E. D. 03 • 1. Opaque sets of degree . Amer. Math. Monthly 71 (1964) 535- 537.

Hildebrand, S. K. 03 • 1. A connected topology for the unit interval. Fund. Math. 61 (1967) 133-140. 2. An interesting metric space. Math. Mag. 41 (1968) 244-247. (with Milnes, Harold Willis) 3. The separation axioms for invertible spaces. Amer. Math. Monthly 75 (1968) 391-392.(with Poe, R. L.) 4. Connectivity functions and retracts. Fund. Math. 57 (1965) 237-245.(with Sanderson, D. E.)

Sims B. T. 03 •

1. Between T2 and T3. Math. Mag. 40 (1967) 25-26.

Irudayanathan, A. 03 • 334 CHAPTER 9

1. Connected-open topology for function spaces. Nederl. Adad. Wetensch. Proc. Ser. A69 = Indag. Math. 28 (1966) 22-24. (with Naimpally, S.)

McCoy, R. A. 03 • 1. Some applications of Henderson’s open embedding theorem of F -manifolds. Compositio Math. 21 (1969) 295-298.

Curtis D. W. 03 • 1. Relatively contractive relations in pairs of generalized uniform spaces. J. London Math. Soc. 44 (1969) 100-106.(with Mathews, J. C.) 2. Generalized uniformities for pairs of spaces. Topology Confer- ence (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968, 212-246.

Goblirsch, Richard Paul 02 • 1. An area for simple surfaces. Annals. of Math. (2) 68 (1968) 231246. 2. On decompositions of 3-space by linkages. Proc. Amer. Math. Soc. 10 (1959) 728-730.

Lehner, Guydo Rene 02 • 1. Extending homeomorphisms on the pseudo-arc. Trans. Amer. Math. Soc. 98 (1961) 369-394.

Bean, Ralph 03 • 1. Disks in E3. I. Subsets of disks having neighborhoods lying on 2spheres. Trans. Amer. Math. Soc. 112 (1964) 206-213. 2. Disks in E3. II. Disks which ”almost” lie in a 2-sphere. Trans. Amer. Math. Soc. 119 (1965) 123-124. 3. Decompositions of E3 with a null sequence of starlike equivalent nondegenerate elements are E3. Illinois J. Math. 11 (1967) 21-23. 4. Decompositions of E3 which yield E3. Paciﬁc J. Math. 20 (1967) 411413. ACADEMIC DESCENDANTS AND PUBLICATIONS 335

5. Repairing embeddings and decompositions in S3. Duke Math. J. 36 (1969) 379-385.

Bailey, John L. 04 • 1. Point-like upper semi-continuous decompositions of S3. Illinois J. Math. 13 (1969) 674-679.

Hyman, D. M. 03 • 1. A generalization of the Borsuk-Whitehead-Eanner theorem. Pa- ciﬁc J. Math. 23 (1967) 263-271. 2. ANR divisors and absolute neighborhood contractibility. Fund. Math. 62 (1968) 61-73. 3. A category slightly larger than the metric and CW=categaries. Mich. Math. J. 15 (1968) 193-214. 4. A note on closed maps and metrizability. Proc. Amer. Math. Soc. 21 (1969) 109-112. 5. On decreasing sequences of compact absolute retracts. Fund. Math. 64 (1969) 91-97.

Brown, Morton 02 • 1. Weak n-homogeneity implies weak (n l) homogeneity. Proc. Amer. Math. Soc. 10 (1959) 644-647. − 2. A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc. 66 (1960) 74-76. 3. Some applications of an approximation theorem for inverse limits. Proc. Amer. Math. Soc. 11 (1960) 478-483. 4. On the inverse limit of Euclidean N-spheres. Trans. Amer. Math. Soc. 96 (1960) 129-134. 5. The monotone union of open n-cells is an open n-cell. Proc. Amer. Math. Soc. 12 (1961) 812-814. 6. Locally flat embeddings of topological manifolds. Topology of 3manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N. J., 1962, 92- 94. 7. A mapping theorem for untriangulated manifolds. Topology of 3-manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N. J., 1962, 92- 94. 336 CHAPTER 9

8. Locally ﬂat embeddings of topological manifolds. Annals. of Math. (2) 75 (1962) 331-341. 9. On a theorem of Fisher concerning the homeomorphism group of a manifold. Mich. Math. J. (1962) 403-405. 10. A note on Kister’s isotopy. Mich. Math. J. 14 (1967) 95-96. 11. Wild cells and spheres in higher dimensions. Mich. Math. J. 14 (1967) 219-224. 12. Pushing graphs around. Proc. of the Conference on the topology of manifolds (Michigan State Univ. E. Lansing, Mich., 1967) Prindle, Weber, & Schmidt, Boston, Mass., 1968, 19-22. 13. A note on Cartesian products. Amer. J. Math. 91 (1969) 32-36. 14. Stable structures on manifolds. Bull. Amer. Math. Soc. 69 (1963) 51-58.(with Gluck, Herman) 15. Stable structures on manifolds. I. Homeomorphisms of Sn . An- nals. of Math. (2) 79 (1964) 1-17. 16. Stable structures on manifolds. II. Stable manifolds. Annals. of Math. (2) 79 (1964) 18-44. 17. Stable structures on manifolds. III. Applications. Annals. of Math. (2) 79 (1964) 45-58.

Kister, J. M. 02 • 1. Isotopies in 3-manifolds. Trans. Amer. Math. Soc. 97 (1960) 213224. 2. A theorem on infinite regular neighborhoods and an application to periodic maps on E . Topology of 3-manifolds and related topics (Proc. the Univ. Georgia Institute, 1961) 221-222, Prentice- Hall, Englewood Cliffs, N. J., 1962. 3. Questions on isotopies in manifolds. Topology of 3-manifolds and related topics (Proc. the Univ. of Georgia Institute, 1961) 229-230. Prentic-Hall, Englewood Cliffs, N. J. 1962. 4. Examples of periodic maps on Euclidean spaces without fixed points. Bull. Amer. Math. Soc. 67 (1961) 471-474. 5. Uniform continuity and compactness in topological groups. Proc. Amer. Math. Soc. 13 (1962) 37-40. 6. Differentiable periodic actions on E without fixed points. Amer. J. Math. 85 (1963) 316-319. ACADEMIC DESCENDANTS AND PUBLICATIONS 337

7. Microbundles are ﬁbre bundles. Bull. Amer. Math. Soc. 69 (1963) 854-857. 8. Homotopy types of ANR’s. Proc. Amer. Math. Soc. 19 (1968) 195. , and Bing, R. H. 9. Taming complexes in hyperplanes. Duke Math. J. 31 (1964) 491-511. 10. Microbundles are ﬁbre bundles. Annals. of Math. (2) 80 (1964) 190-199 . 11. Inverses of Euclidean bundles. Mich. Math. J. 14 (1967) 349- 352. 12. Equivalent imbeddings of compact abelian Lie grops of transformations. Math. Ann. 148 (1962) 89-93.(with Mann, L. N.) 13. Isotopy structure of compact Lie groups on complexes. Mich. Math. J. 9 (1962) 93-96. 14. Locally Euclidean factors of E4 which cannot be imbedded in E3. Annals. of Math. (2) 76 (1962) 541-546.(with McMillan, D. R., Jr.)

Rosen R H 02 • 1. Fixed points for multi-valued functions on snake-like continua. Proc. Amer. Math. Soc. 10 (1959) 167-173. 2. Decomposing 3-space into circles and points. Proc. Amer. Math. Soc. 11 (1960) 918-928. 3. E4 is the Cartesian product of a totally non-Euclidean space and E1. Annals. of Math. (2) 73 (1961) 349-361. 4. Examples of non-orthogonal involutions of Euclidean spaces. An- nals. of Math. (2) 78 (1963) 560-566. 5. Polyhedral neighborhoods in triangulated manifolds. Bull. Amer. Math. Soc. 69 (1963) 359-361. 6. Stellar neighborhoods in polyhedral manifolds. Proc. Amer. Math. Soc. 14 (1963) 401-406. 7. The five dimensional polyhedral Schoenflies theorem. Bull. Amer. Math. Soc. 70 (1964) 511-516. 8. Addendum to a paper on fixed points. Yokohama Math. J. 16 (1968) 9-10. 9. A note concerning certain subcomplexes of triangulated manifolds. Yokohama Math. J. 16 (1968) 5-7. 338 CHAPTER 9

McMillan, D. R., Jr. 02 • 1. Cartesian products of contractible open manifolds. Bull. Amer. Math. Soc. 67 (1961) 510-514. 2. On homologically trivial 3-manifolds. Trans. Amer. Math. Soc. 98 (1961) 350-367. 3. Summary of results on contractible open manifolds. Topology of 3-manifolds and related topics (Proc. the Univ. of Georgia Institute, 1961) 100-102. Prentice-Hall, Englewood Cliﬀs, N. J., 1962. 4. Some contractible open 3-manifolds. Trans. Amer. Math. Soc. 102 (1962) 373-382. 5. Homeomorphisms on a solid torus. Proc. Amer. Math. Soc. 14 (1963) 386-390. 6. A criterion for cellularity in a manifold. Annals. of Math. (2) 79 1964) 327-337. 7. Taming Cantor sets in En. Bull. Amer. Math. Soc. 70 (1964) 706708. 8. The singular points of a topological embeddings. Duke Math. J. 31 (1964) 711-716. 9. Local properties of the embedding of a graph in a three-manifold. Canadian J. Math. 18 (1966) 517-528. 10. A criterion for celluarity in a manifold. II. Trans Amer. Math. Soc. 126 (1967) 217-224. 11. Neighborhoods of surfaces in 3-manifolds. Mich. Math. J. 14 (1967) 161-170. 12. Some topological properties of piercing points. Paciﬁc J. Math. 22 (1967) 313-322. 13. Piercing a disk along a cellular set. Proc. Amer. Math. Soc. 19 (1968) 153-157. 14. Strong homotopy equivalence of 3-manifolds. Bull. Amer. Math. Soc. 73 (1967) 718-722. 15. Compact, acyclic subsets of three-manifolds. Mich. Math. J. 16 (1969) 129-136. 16. Cellularity of sets in products. Mich. Math. J. 9 (1962) 299- 302.(with Curtis, M. L.) 17. Locally nice embeddings of manifolds. Amer. J. Math. 8 (1966) 1-19.(with Hempel, J. P.) ACADEMIC DESCENDANTS AND PUBLICATIONS 339

18. Covering three-manifolds with open cells. Fund. Math. 64 (1969) 99-104.(with Hempel, J. P.) 19. Locally Euclidean factors of E4 which cannot be imbedded in E3. Ann of Math. (2) 76 (1962) 541-546.(with Kister, J. M.) 20. Tangled embeddings of one-dimensional continua. Proc. Amer. Math. Soc. 22 (1969) 378-385.(with Row, W. H.) 21. On contractible open manifolds. Proc. Cambridge Philos. Soc. 58 (1962) 221-224.(with Zeeman, E. C.)

Jaco, William 03 • 1. Constructing 3-manifolds from group homomorphisms. Bull. Amer. Math. Soc. 74 (1968) 936-940. 2. Three-manifolds with fundamental group a free product. Bull. Amer. Math. Soc. 75 (1969) 972-977. 3. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 (1969) 365-379.

Evans. B. D. 04 • 1. Convex components, extreme points, and the convex kernel. Proc. Amer. Math. Soc. 21 (1969) 83-87.

Row W. H. 03 • 1. Tangled embeddings of one-dimensional continua. Proc. Amer. Math. Soc. 22 (1969) 378-385.(with McMillan, D. R., Jr.)

Gillman David S. 02 • 1. Tame subsets of 2-spheres in E3. Topology of 3-manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) 26-28. Prentice-Hall, Englewood Cliﬀs, N. J., 1962. 2. Side approximatian, missing an arc. Amer. J. Math. 85 (1963) 459-476. 3. Note concerning a wild sphere of Bing. Duke Math. J. 31 (1964) 247-254. 4. Sequentially l-ULC tori. Trans. Amer.-Math. Soc. 111 (1964) 449456. 5. Unknotting 2-manifolds in 3-hyperplanes of E4. Duke Math. J. 33 (1966) 229-245. 340 CHAPTER 9

6. The spinning and twisting of a complex in a hyperplane. Annals of Math. (2) 85 (1967) 32-41. 7. Free curves in E3. Paciﬁc J. Math. 28 (1969) 533-542.

Hempel, John P. 02 • 1. Construction of orientable 3-manifolds. Topology of 3-manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) 207-212. Prentice-Hall, Englewood Cliﬀs, N. J., 1962. 2. A simply connected 3-manifold in S3 if it is the sum of a solid torus and the complement of a torus knot. Proc. Amer. Math. Soc. 15 (1964) 154-158. 3. A surface in S3 is tame if it can be deformed into each complementary domain. Trans. Amer. Math. Soc. 111 (1964) 273-287. 4. Free surfaces in S3. Trans. Amer. Math. Soc. 141 (1969) 263-270. 5. Locally nice embeddings of manifolds. Amer. J. Math. 88 (1966) 1-19 .(with McMillan, D. R., Jr.) 6. Covering three-manifolds with open cells. Fund. Math. 64 (1969) 99-104. 7. Quantum tunnel model for hard spheres. J. Chem. Phys. 41 (1964) 3487-3490.(with Oden, Lyn)

Heil, Wolfgang 03 • 1. On p2 irreducible 3-manifolds. Bull. Amer. Math. Soc. 75 (1969) 772-775. 2. On the existence of incompressible surfaces in certain 3-manifolds. Proc. Amer. Math. Soc. 23 (1969) 704-707.

Casler, Burtis G. 02 • 1. An embedding theorem for connected 3-manifolds with boundary. Proc. Amer. Math. Soc. 16 (1965) 559-566. 2. On the sum of two solid Alexander horned spheres. Trans. Amer. Math. Soc. 116 (1965) 135-150. 3. The generalized Poincare conjecture for weak-star 5-manifolds. Topology 6 (1967) 69-75.

Glaser, L. C. 02 • ACADEMIC DESCENDANTS AND PUBLICATIONS 341

1. Intersections of combinatorial balls and of Euclidean spaces. Bull. Amer. Math. Soc. 72 (1966) 68-71. 2. Contractible complexes in Sn. Proc. Amer. Math. Soc. 16 (1965) 1357-1364. 3. Dimension lowering monotone non-compact mappings of En. Fund. Math. 58 (1966) 171-181. 4. 1-1 continuous mappings onto En. Amer. J. Math. 88 (1966) 237243. 5. Intersections of combinatorial balls and of Euclidean spaces. Trans. Amer. Math. Soc. 122 (1966) 311-320. 6. Uncountably many contractible open 4-manifolds. Topology 6 (1966) 37-42. 7. Bing’s house with two rooms from 1-1 continuous map onto E3: Amer. Math. Monthly 74 (1967) 156-160. 8. On the double suspension of certain homotopy 3-spheres. An- nals. of Math. (2) 85 (1967) 494-507. 9. Uncountably many almost polyhedral wild (k 2)-cells in Ek for k > 4. Paciﬁc J. Math. 27 (1968) 267-273. − 10. Monotone noncompact mapping of Er onto Ek for r < 4 and k > 4. Proc. Amer. Math. Soc. 23 (1969) 282-286. 11. On a conjecture related to the suspension of homotopy 3-spheres and fake cubes. Mich. Math. J. 15 (1968) 325-338.

Hosay, Norman 02 • 1. A proof of the slicing theorem for 2-spheres. Bull. Amer. Math. Soc. 75 (1969) 370-374.

Price, T. M. 02 • 1. A necessary condition that a cellular upper semi-continuous decomposition of En yield En. Trans. Amer. Math. Soc. 122 (1966) 427-435. 2. Equivalence of embeddings of k-complexes in Ek for n < 2k + 1. Mich. Math. J. 13 (1966) 65-69. 3. A knotted cell pair with knot group Z. Illinois J. Math. 12 1968) 201-204. 4. Decompositions of S3 and pseudo-isotopies. Trans. Amer. Math. Soc. 140 (1969) 295-299. 342 CHAPTER 9

5. Decompositions into compact sets with W properties. Trans. Amer. Math. Soc. 141 (1969) 433-442.(with Armentrout, Steve) 6. Unknotting locally ﬂat cell pairs. Illinois J. Math. 10 (1966) 425-430.(with Glaser, L. C.)

Nicholson, Victor 03 • 1. Mapping cylinder neighborhoods. Trans. Amer. Math. Soc. 143 (1969) 259-268.

Henderson, David W. 02 • 1. A short proof of Wedderburn’s theorem. Amer. Math. Monthly 72 (1965) 385-386. 2. Extensions of Dehn’s lemma and the loop theorem. Trans. Amer. Math. Soc. 120 (1965) 448-469. 3. Self-unlinked simple closed curves. Trans. Amer. Math. Soc. 120 (1965) 470-480. 4. Relative general position. Pacific J. Math. 18 (1966) 513-523. 5. An infinite-dimensional compactum with no positive-dimensional compact subsets - a simpler construction. Amer. J. Math. 89 (1967) 105-121. 6. Each strongly infinite-dimensional compactum contains a hereditarily infinite-dimensional compact subset. Amer. J. Math. 89 (1967) 122-123. 7. D-dimension. I. A new transfinite dimension. Pacific J. Math. 26 (1968) 109-113. 8. D-dimension II. Separable spaces and compactifications. Pacific J. Math. 26 (1968) 109-113. 9. A lower bound for transfinite dimension. Fund. Math. 63 (1968) 167-173. 10. Infinite-dimensional manifolds are open subsets of Hilbert spaces. Bull. Amer. Math. Soc. 75 (1969) 759-762. 11. Infinite-dimensional manifolds are open subsets of Hilbert spaces. Topology 9 (1969) 25-33. 12. Open subsets of Hilbert spaces. Compositio Math. 21 (1969) 312318. ACADEMIC DESCENDANTS AND PUBLICATIONS 343

13. Negligible subsets of infinite-dimensional manifolds. Compositio Math. 21 (1969) 143-150.(with Anderson, R. D.; West, James E.) 14. Another generalization of Brouwer’s fixed point theorem. Proc. Amer. Math. Soc. 19 (1968) 176-177.(with Livesay, C. G.) 15. Topological classification of infinite dimensional manifolds by homotopy type. Bull. Amer. Math. Soc. 76 (1970) 121-124.(with Schori, R.)

Cutler, William H. 03 • 1. Negligible subsets of inﬁnite-dimensional Frechet manifolds. Proc. Amer. Math. Soc. 23 (1969) 668-675.

Dancis Jerome 02 • 1. Some nice embeddings in the trivial range. Topology seminar (Wisconsin, 1965) 159-170. Annals. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N. J., 1966. 2. Approximations and isotopies in the trivial range. Topology Seminar (Wisconsin, 1965) 171-187. Annals. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N. J., 1966. 3. Topological analogues of combinatorial techniques. Proc. of the Conference on the topology of manifolds. (Michigan State Univ., E. Lansing, Michigan, 1967) 31-46. Prindle, Weber, & Schmidt, Boston, Mass., 1968.

Cobb, John I., III 02 • 1. On ordering inﬁnitely many small homeomorphisms. Proc. Amer. Math. Soc. 23 (1969) 64-67.

McAllister, B. L. 02 • 1. Cyclic elements in topology, a history. Amer. Math. Monthly 731 (1966) 337-350. 2. A note on irreducible separation. Fund. Math. 63 (1968) 143- 144. 3. Binary relations on Boolean matrices. Math. Mag. 43 (1970) 8-14.(with Feichtinger, Oskar) 4. A new ”cyclic” element theory. Math Zeit. 101 (1967) 157-164. (with McAuley, L. F.) 344 CHAPTER 9

Craggs, Robert F. 02 • 1. Improving the intersection of polyhedra in 3-manifolds. Illinois J. Math. 12 (1968) 567-586. 2. Building Cartesian products of surfaces with [0,1]. Trans. Amer. Math. Soc. 144 (1969) 391-425.

Jones, S. L. 02 • 1. The impossibility of ﬁlling E with arcs. Bull. Amer. Math. Soc. 74 (1968) 155-159.

Yohe J. M. 02 • 1. Structure of hereditarily inﬁnite dimensional spaces. Proc. Amer. Math. Soc. 20 (1969) 179-184. 2. Monotone mapping properties of hereditarily inﬁnite dimensional spaces. Proc. Amer. Math. Soc. 22 (1969) 639-645.

Wright, Perrin 02 • 1. A uniform generalized Schoenﬂies theorem. Bull. Amer. Math. Soc. 74 (1968) 718-721. 2. A uniform generalized Schoenﬂies theorem. Annals. of Math. (2) 89 (1969) 292-304.

Daverman R. J. 02 • 1. A new proof for the Hosay-Lininger theorem about crumpled cubes. Proc. Amer. Math. Soc. 23 (1969) 52-54. 2. A dense set of sewings of two crumpled cubes yields S3. Fund. Math. 65 (1969) 51-60.(with Eaton, William T.) 3. An equivalence for the embeddings of cells in a 3-manifold. Trans. Amer. Math. Soc. 145 (1969) 369-381. 4. Remarks on the Claytor imbedding theorem- Duke Math. J. 19 (1952) 199-202.

5. A remark on ∗-spaces. Mich. Math. J. 1 (1952) 79-80. L 6. Aﬃne structures in 3-manifolds. VI. Compact spaces covered by two Euclidean neighborhoods. Annals. of Math. (2) 58 (1953) 107. ACADEMIC DESCENDANTS AND PUBLICATIONS 345

7. Affine structures in 3-manifolds. VII. Disks which are pierced by intervals. Annals. of Math. (2) 58 (1953) 403-408. 8. The use of induction in existence proofs. Amer. Math. Monthly 61 192-193 (1954) 15-494. 9. Affine structures in 3-manifolds. VIII. Invariance of the knot- types; local tame imbedding. Annals. of Math. (2) S9 (1954) 159-170. 10. Simply connected 3-manifolds. Topology of 3-manifolds and related topics (Proc. the Univ. of Georgia Institute, 1961) 196-197. Prentice-Hall, Englewood Cliffs, N. J., 1962. 11. Periodic homeomorphisms of the 3-sphere. Illinois J. Math. 6 (1962) 206-225. 12. Elementary geometry from an advanced standpoint. Addison- Wesley Publishing Co., Inc., Reading, Mass. – Palo Alto, Calif. -London, 1963. x+419. 13. A monotonic mapping theorem for simply connected 3-manifolds. Illinois J. Math. 12 (1968) 451-474. 14. Almost locally polyhedral spheres. Annals. of Math. 57 (1953) 575-578.(with Harrold, O. G.)

Bing, R. H. 01 • 1. Extending monotone decompositions of 3-manifolds. Trans. Amer. Math. Soc. 149 (1970) 351-369. 2. The monotone mapping problem. Topology of Manifolds. (Proc. Inst. Univ. of Georgia, Athens, Georgia, 1969) Markham, Chicago, Ill., 1970. 3. Cubes with knotted holes. Trans. Amer. Math. Soc. 155 (1971) 217-231.(with Martin, J. M.) 4. Monotone images of E3. Proc. First Conf. on Monotone Map- pings and Open Mappings. (SUNY, Binghamton, Binghmaton, N. Y., 1970) 55-77. State University of New York at Bingham- ton, Binghamton, New York, 1971.

Sanderson, D. E. 02 • 1. An inﬁnite-dimensional Schoenﬂies theorem. Trans. Amer. Math. Soc. 148 (1970) 33-39.

Schmidt, D. L. 03 • 346 CHAPTER 9

1. A proof of the Moore metrization theorem. Colloq. Math. 22 (1970) 59-60.(with Rozycki, Eugene P.)

Hildebrand, S. K. 03 • 1. Non-isolated minimizing arcs. SIAM J. Appl. Math. 18 (1970) 139-149.(with Milnes H. W.)

McCoy, R. A. 03 • 1. Annulus conjecture and stability of the homeomorphisms in in- ﬁnitedimensional normed linear spaces. Proc. Amer. Math. Soc. 24 (1970) 272-277. 2. On boundedly metacompact and boundedly paracompact spaces. Proc. Amer. Math. Soc. 25 (1970) 335-342.(with Fletcher P.; Slover, R.)

Curtis, D. W. 03 •

1. Property Z for function-graphs and ﬁnite-dimensional sets in I∞ and S. Composito Math. 22 (1970? 19-22.

Bean, Ralph 03 • 1. Extending monotone decompositions of 2-spheres to trivial decompositions of E3. Duke Math. J. 38 (1971) 539-544. 2. Decompositions of E3 which satisfy a uniform Lipschitz condition are factors of E4. Fund. Math. 70 (1971) no. 2, 109-llS.

Bailey, J. L. 04 • 1. A class of decompositions of En which are factors of En+1 Trans. Amer. Math. Soc. 148 (1970) 561-575.

Kister, James 02 • 1. Counting topological manifolds. Topology 9(1970) 149-151. (with Cheeger, J.)

Miller R. T. 03 • 1. Close isotopies on pointwise linear manifolds. Trans. Amer. Math. Soc. 151 (1970) 597-628. ACADEMIC DESCENDANTS AND PUBLICATIONS 347

Edwards, Richard 03 • 1. A suﬃcient condition that the limit of a sequence of continuous functions be an embedding. Proc. Amer. Math. Soc. 26 (1970) 224-225.

Connelly, Robert 03 • 1. A new proof of Brown’s collaring theorem. Proc. Amer. Math. Soc. 27 (1971) 180-182. 2. Unknotting close polyhedra in codimension three. Topology of Manifolds. (Proc. Inst. Univ. of Georgia, Athen, Georgia, 1969) 384-388. Markham, Chicago, Illinois, 1970.

Rosen, Ronald H. 02 • 1. Concerning suspension spheres. Proc. Amer. Math. Soc. 23 (1969) 225-231.

McMillan, D. R. 02 • 1. Retracting three-manifolds onto ﬁnite graphs. Illinois J. Math. 14 (1970) 150-158.(with Jaco, William) 2. Acyclicity in three-manifolds. Bull. Amer. Math. Soc. 76 (1970) 942-964. 3. Boundary-preserving mappings of a 3-manifold. Topology of Manifolds. (Proc. Inst. Univ. of Georgia, Athens, Georgia, 1969) 161-175. Markham, Chicago, Illinois, 1970.

Jaco, William N. 03 • 1. On certain subgroups of the fundamental group of a closed surface. Proc. Cambridge Philos. Soc. 67 (1970) 17-18. 2. Retracting three-manifolds onto ﬁnite graphs. Illinois J. Math. 14 (1970) 150-158.(with McMillan, D. R.) 3. Surfaces embeddable in M 3 S1 Canad.J. Math. 22 (1970) 553- 558. × 4. Stable equivalence of splitting homeomorphisms. Topology of Manifolds. (Proc. Inst. Univ. of Georgia, Athens, Georgia, 1969) Markham, Chicago, Illinois, 1970. 153-156.

Lyon, Herbert C. 04 • 348 CHAPTER 9

1. Incompressible surfaces in knot spaces. Trans. Amer. Math. Soc. 157 (1971) 53-62.

Wright, Alden H. 03 • 1. Mapping cylinders and 4-manifolds. Topology of Manifolds. (Proc. Inst. Univ. of Georgia, Athens, Georgia, 1969) 424-427. Markham, Chicago, Illinois, 1970.(with Lacher, R. C.)

Heil, Wolfgang 03 • 1. On the existence of incomprehensible surfaces in certain 3-manifolds. II Proc. Amer. Math. Soc. 25 (1970) 429-432.

Glaser, Leslie 02 • 1. Euclidean (q+r) space modulo an r-plane of collapsible p-complexes. Trans. Amer. Math. Soc. 157 (1971) 261-278.

Price, T. M. 02 • 1. Finding a boundary for a 3-manifold. Annals. of Math. (2) 91 (1970) 223-235.(with Husch, L. S.) 2. A class of embeddings of Sn 1 and Bn in Rn. Proc. Amer. Math. Soc. 29 (1971) 208-210.(with Cantrell, J. C.; Rushing, T. B.)

Geoghagen, Ross 03 • 1. Manifolds of piecewise linear maps and a related normed linear space. Bull. Amer. Math. Soc. 77 (1971) 629-632.

Dancis Jerome 02 • 1. PL approximations and embeddings of manifold in the 4 range. Topology of Manifolds. (Proc. Inst. Univ. of Georgia, Athens, Georgia, 1969) 335-340. Markham, Chicago, Illinois, 1970.(with Berkowitz, Harry W.) 2. PL approximations of embeddings and isotopes of polyhedra in the metastable range. ibid. 341-352. 3. An introduction to embedding spaces for geometrical topologists. ibid. 389-393.

Craggs, Robert 02 • ACADEMIC DESCENDANTS AND PUBLICATIONS 349

1. Involutions of the 3-sphere which ﬁx 2-spheres. Paciﬁc J. Math. 32 (1970) 307-321. 2. Small ambient isotopies of a 3-manifold which transform one embedding of a polyhedron into another. Fund. Math. 68 (1970) 225-256.

Jones, S. L. 02 • 1. Continuous collections of compact manifolds. Duke Math. J. 37 (1970) 579-587. 2. Degree one mappings on three-manifolds. Proc. First Conf. on Monotone r Mappings and Open Mappings. (SUNY at Bing- hamton, Binghamton, N.Y., 1970) 78-86. State Univ. of New York at Binghamton, Binghamton, N. Y., 1971.

Wright, Perrin T. 02 • 1. Collapsing K I to vertical segments. Proc. Cambridge Philos. Soc. 69 (1971)× 71-74. 2. Radial engulﬁng in codimension three. Duke Math. J. 38 (1971) 295-298. n 1 3. Covering isotopies of M − in N . Proc. Amer. Math. Soc. 29 (1971) 595-598.

Daverman Robert 02 • 1. A self-universal crumpled cube which is not universal. Bull. Amer. Math. Soc. 76 (1970) 740-742.(with Bass, Charles D.) 2. Non-homeomorphic approximations of manifolds with surfaces of bounded genus. Duke Math. J. 37 (1970) 619-625. 3. Sewings of crumpled cubes. Topology of Manifolds. (Proc. Inst., Univ. Georia, Athens, Georgia, 1969) 124-128. Markham, Chicago, Illinois, 1970. 4. On the number of non-piercing points in certain crumpled cubes. Paciﬁc J. Math. 34 (1970) 33-43.

Bass, Charles D. 03 • 1. A self-universal crumpled cube which is not universal. Bull. Amer. Math. Soc. 76 (1970) 740-742.(with Daverman, R. J.)

Jensen, Richard A. 02 • 350 CHAPTER 9

1. Cross sectionally connected 2-spheres are tame. Bull. Amer. Math. Soc. 76 (1970) 1036-1038.

Olinik, M. 02 • 1. Factoring monotone maps of En. Topology of Manifolds. (Proc. Inst. Univ. of Georgia, Athens, Georgia, 1969) 185-189. Markham, Chicago, Illinois, 1970. ACADEMIC DESCENDANTS AND PUBLICATIONS 351 E. E. Moise and his Mathematical Descendandts

01 Moise, E. E. 02 Munkres, James R., University of Michigan, 1956 02 Finney, Ross Lee, University of Michigan, 1962

Publications of E. E. Moise and his Mathematical Descendants

Moise, E. E. 01 • 1. An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua. Trans. Amer. Math. Soc. 63 (1948) 581-594. 2. A theorem on monotone interior transformations. Bull. Amer. Math. Soc. 55 (1949) 810-811. 3. Grille decomposition and convexification theorems for compact metric locally connected continua. Bull. Amer. Math. Soc. 55 (1949) 1111-1121 4. A note on the pseudo-arc. Trans. Amer. Math. Soc. 67 (1949) 5758. 5. Affine structures in 3-manifolds. I. Polyhedral approximations of solids. Annals of Math.. (2) 54 (1951) 506-533. 6. A note of correction. Proc. Amer. Math. Soc. 2 (1951) 838. 13-265. 7. Affine structures in 3-manifolds. II. Positional properties of 2- spheres. Annals of Math.. (2) 55 (1952) 172-176. 8. Affine structures in 3-manifolds. III. Tubular neighborhoods of linear graphs. Annals of Math.. (2) 55 (1952) 203-214. 9. Affine structures in 3-manifolds. IV. Piecewise linear approximations of homeomorphisms. Annals of Math.. (2) 55 (1952) 215-222. 10. Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Annals of Math. (2) 56 (1952) 96-114. 352 CHAPTER 9

Munkres, J. R. 02 • 1. Concordance inertia groups. Adv. in Math. 4 (1970) 224-235 . 2. Concordance classes of bundles over spheres. Mich. Math. J. 17 (1970) 57-63.

Finney, R. L. 02 • 1. An elementary proof of a theorem of Bkouche: Papers from the ”Open House for Algebraists (Aarhus, 1970) 109-111. Mat. Inst. Aarhus, Univ. Aarhus, 1970.(with Rotman, Joseph) ACADEMIC DESCENDANTS AND PUBLICATIONS 353 R. D. Anderson and his Mathematical Descendandts

01 Anderson, R. D. 02 Fisher, Gordon McCrea, Louisiana State University, 1960 02 Nunnally, Ellard, Louisiana State University, 1964 02 Vobach, Arnold R., Louisiana State University, 1964 03 Riecke, C. V., University of Houston, 1971 03 O’Steen,02 Brechner, David, University Beverly of L., Houston, Louisiana 1972 State University, 1965 02 Wong, Raymond Yen-Tin, Louisiana State University, 1967 02 West, James EdWard, Louisiana State University, 1968 02 Chapman. Thomas A., Louisiana State University, 1970

Publications of R. D. Anderson and his Mathematical Descendants

Anderson, R. D. 01 • 1. On the application of quantum mechanics to mortality tables. J. Inst. Actuar. 71 (1942) 228-258. 2. Concerning upper semi-continuous collections of continua. Trans. Amer. Math. Soc. 67 (1949) 451-460. 3. Monotone interior dimension-raising mappings. Duke Math. J. 19 (1952) 359-366. 4. Continuous collections of continuous curves in the plane. Proc. Amer. Math. Soc. 3 (1952) 647-657. 5. On monotone interior mappings in the plane. Trans. Amer. Math. Soc. (1952) 211-222. 6. Continuous collections of continuous curves. Duke Math. J. 21 (1954) 363-367. 7. Some remarks on totally disconnected sections of monotone open mappings. Bull. Acad. Polon. Sci. Cl. III. 4 (1956) 329-330. 8. Atomic decompositions of continua. Duke Math. J. 23 (1956) 507-514. 9. Open mappings of compact continua. Proc. Nat. Acad. Sci. 42 (i556) 347-349. 354 CHAPTER 9

10. One-dimensional continuous curves. Proc. Nat. Acad. Sci. (1956) 760-762. 11. Zero-dimensional compact groups of homeomorphisms. Pacific J. Math. 7 (1957) 797-810. 12. The algebraic simplicity of certain groups of homeomorphisms. Amer. J. Math. 80 (1958) 955-963. 13. A characterization of the universal curve and a proof of its homo geneity. Annals of Math. (2) 67 (1958) 313-324. 14. One-dimensional continuous curves and a homogeneity theorem. Annals of Math. (2) 68 (1958) 1-16. 15. Homeomorphisms of 2-dimensional continua. Topology of 3- manifolds and related topics. (Proc. the Univ. of Georgia In- stitue, 1961) 41. Prentice-Hall, Englewood Cliffs, N. J., 1962. 16. On homeomorphisms as products of conjugates of a given homeomorphism and its inverse. Topology of 3-manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) 231-234. PrenticeHall, Englewood Cliffs, N. J., 1962. 17. On raising flows and mappings. Bull. Amer. Math. Soc. 69 (1963) 259-264. 18. Homeomorphisms of 2-dimensional continua. General Topology and its Relation to Modern Analysis and Algebra (Proc. Sym- pos., . Prague, 1961) 55-58. Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962. 19. Quasi-universal flows and semi-flows. Fund. Math. 57 (1965) 1-8. 20. Hilbert space is homeomorphic to the countable infinite product of lines. Bull. Amer. Math. Soc. 72 (1966) 515-519. 21. On a theorem of Klee. Proc. Amer. Math. Soc. 17 (1966) 14011404. 22. Topological properties of the Hilbert cube and the infinte product of open intervals. Trans. Amer. Math. Soc. 126 (1967) 200-216. 23. On topological infinite deficiency. Mich. Math. J. 14 (1967) 365-383. 24. Universal and quasi-universal flows. Topological Dynamics (Proc. of the Symposium at Colorado State Univ., Fort Collins, Colo., 1967) 1-16. Benjamin, New York, 1968. ACADEMIC DESCENDANTS AND PUBLICATIONS 355

25. Strongly negligible sets in Frechet manifolds. Bull. Amer. Math. Soc. 75 (1969) 64-67. 26. A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines. Bull. Amer. Math. Soc. 74 (1968) 771-792.(with Bing, R. H.) 27. A plane continuum no two of whose nondegenerate subcontinua are homeomorphic: An application of inverse limits. Proc. Amer. Math. Soc. 10 (1959) 347-353.(with Choquet, Gustave) 28. Concerning closures of images on the reals. Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 9 (1961) 807-810.(with Connell, E. H.) 29. Closures of images on the reals. Topology of 3-manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) 45. Prentice-Hall, Englewood Cliffs, N. J., 1962 30. A note on continuous collections of continuous curves filling up a continuous curve in the plane. Proc. Amer. Math. Soc. 5 (1954) 748-752.(with Hamstrom, Mary Elizabeth) 31. On spaces filled up by continuous collections of atriodic continuous curves. Proc. Amer. Math. Soc. 6 (1955) 766-769. 32. Negligible subsets of infinite-dimensional manifolds. Compositio Math. 21 (1969) 143-150.(with Henderson, David W.; West, James E.) 33. An example in dimension theory. Proc. Amer. Math. Soc. 18 (1967) 709-713.(with Keisler, J. E.) 34. Convex functions and upper semicontinuous collections. Duke Math. J. 19 (1952) 349-357.(with Klee, V. L., Jr.) 35. On extending homeomorphisms to Frechet manifolds. Proc. Amer. Math. Soc. 25 (1970) 283-289.(with McCharen, John D.) 36. A factor theorem for Frechet manifolds. Bull. Amer. Math. Soc. 75 (1969) 53-56.(with Schori, R. M.) 37. Factors of infinite-dimensional manifolds. Trans. Amer. Math. Soc. 142 (1969) 315-330.

Fisher, Gordon McCrea 02 • 1. On the group ofall homeomorphisms of a manifold. Trans. Amer. Math. Soc. 97 (1960) 193-212.

Nunnally, Ellard V. 02 • 356 CHAPTER 9

1. Dilations on invertible spaces. Trans. Amer. Math. Soc. 123 (1966) 437-448. 2. There is no universal-projecting homeomorphism of the Cantor set. Colloq. Math. 17 (1967) 51-52. 3. A factorization of stable homeomorphisims of En. Proc. Amer. Math. Soc. 19 (1968) 387-389.

Vobach Arnold R. 02 • 1. On subgroups of the homeomorphism group of the Cantor set. Fund. Math. 60 (1967) 47-52. 2. A theorem on homeomorphism groups and inverse limits. Nieuw Arch. Wisk. (3) 15 (1967) 31-33. 3. Continua structured by families of simple closed curves. I, II. Czech. Math. J. 18 (93) (1968) 195-210. ibid. 18 (93) (1968) 211-223. 4. Characterizing groups for compact metric spaces. Proc. of the Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State University, Tempe, Arizona, 1968, 293-312. 5. A theorem on homeomorphism groups and products of spaces. Bull. Austral. Math. Soc. 1 (1969) 137-141.

Brechner, Beverly L. 0Z • 1. On the dimensions of certain spaces of homeomorphisms. Trans. Amer. Math. Soc. 121 (1966) 516-548. 2. Homeomorphism groups of dendrons. Paciﬁc J. Math. 28 (1969) 295301. 3. Periodic homeomorphisms on chainable continua. Fund. Math. 64 (1969) 197-202.

Wong, Raymond Yen-Tin 02 • 1. On homeomorphisms of certain inﬁnite dimensional spaces. Trans. Amer. Math. Soc. 128 (1967) 148-154. 2. A wild Cantor set in the Hilbert cube. Paciﬁc J. Math. 24 (1968) 189-193. 3. Extending homeomorphisms by means of collarings. Proc. Amer. Math. Soc. 19 (1968) 1443-1440. ACADEMIC DESCENDANTS AND PUBLICATIONS 357

4. On topological equivalence of n-dimensional linear spaces. Trans. Amer. Math. Soc. 137 (1969) 551-560. 5. Some remarks on hyperspaces. Proc. Amer. Math. Soc. 21 (1969) 600-602. 6. On aﬃne maps and ﬁxed points. J. Math. Anal. Appl. 29 (1970) 158-162.

Anderson, R. D. 01 • 1. On extending homeomorphisms to Frechet-manifolds. Proc. Amer. Math. Soc. 25 (1970) 283-289.(with McCharen, John D.) 2. Apparent boundaries of the Hilbert cube. Proc. Internal. Sym- pos. on Topology and its Applications. (Herceg-Novi, 1968) 60- 66. Savez Drustava Mat. Fiz i Astronom. Belgrade, 1969.

Vobach, A. R. 02 • 1. Two structure theorems for homeomorphism groups. Bull. Aus- tral. Math. Soc. 4 (1971) 63-68.

Brechner, Beverly L. 02 • 1. Strongly locally setwise homogeneous continua and their homeomorphism groups. Trans. Amer. Math. Soc. 154 (1971) 279- 286.

Wong, Raymond Y. T. 02 • 1. Extending homeomorphisms in compactiﬁcation of Frechet spaces. Proc. Amer. Math. Soc. 25 (1970) 548-550. 2. A note on stable homeomorphisms of inﬁnite-dimensional manifolds. Proc. Amer. Math. Soc. 28 (1971) 271-272.

3. Lipshitz conjugation and extension of homeomorphisms in `p- spaces. J. Math. Anal. Appl. 32 (1970) 573-583 4. Stationary isotopies of inﬁnite-dimensional spaces. Trans. Amer. Math. Soc. 156 (1971) 131-136.

West, J. E. 02 • 1. The diﬀeomorphic excision of closed local compacta from in- ﬁnitedimensional Hilbert manifolds. Composito Math. 21 (1969) 271-291. (with Henderson, David W.) 358 CHAPTER 9

2. Triangulated infinite-dimensional manifolds. Bull. Amer. Math. Soc. 76 (1970) 655-660. 3. Infinite products which are Hilbert cubes. Trans. Amer. Math. Soc. 150 (1970) 1-25. 4. The ambient homeomorphy of an incomplete subspace of infinite- dimen5oal Hiert space. Pacific J. Math. 34 tl970) 257-267.

Chapman, T. A. 02 • 1. Inﬁnite deﬁciency in Frechet manifolds. Trans. Amer. Math. Soc. 148 (1970) 137-146.

West, James E. 02 • 1. Extending certain transformation group actions in separable, in- finitedimensional Frechet spaces and the Hilbert cube. Bull. Amer. Math. Soc. 74 (1968) 1015-1019. 2. Fixed-point sets of transformation groups on infinite-product spaces. Proc. Amer. Math. Soc. 21 (1969) 575-582. 3. Fixed point sets of transformation groups on a separable infinite- dimensional Frechet spaces. Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, 446450. 4. Approximating homotopies by isotopies in Frechet manifolds. Bull. Amer. Math. Soc. 75 (1969) 1254-1257. 5. Factoring the Hilbert cube. Bull. Amer. Math. Soc. 76 (1970) 116-120. 6. Negligible subsets of infinite-dimensional manifolds. Composite Math. 21 (1969) 143-150.(with Anderson, R. D.; Henderson, David W.)

Chapman, T. A. 02 • 1. Four classes of separable metric inﬁnite-dimensional manifolds. Bull. Amer. Math. Soc. 76 (1970) 399-403. ACADEMIC DESCENDANTS AND PUBLICATIONS 359 M. Estill Rudin and her Mathematical Descendandts

01 Rudin, M. Estill 02 Tamano, Hisahiro* rank, Univerelty of Wieconein, 1969 02 Warren, Nancy, University of isconsin, 1970

Publications of M. Estill Rudin and her Mathematical Descendants

Rudin, M. Estill 01 • 1. Concerning abstract spaces. Duke Math. J. 17 (1950) 317-327. 2. Separation in non-separable spaces. Duke Math. J. 18 (1951) 623-629. 3. A primitive dispersion set of the plane. Duke Math. J. 19 (1952) 323-328. 4. Concerning a problem of Souslin’s. Duke Math. J. 19 (1952) 629-639. 5. Countable paracompactness and Souslin’s problem. Canadian J. Math. 7 (1955) 543-547. 6. A separable normal nonparacompact space. Proc. Amer. Math. Soc. 7 (1956) 940-941. 7. A subset of the countable ordinals. Amer. Math. Monthly 64 (1957) 351. 8. A topological characterization of sets of real numbers. Paciﬁc J. Math. 7 (1957) 1185-1186. 9. A property of indecomposable connected sets. Proc. Amer. Math. Soc. 8 (1957) 1152-1157. 10. An unshellable triangulation of a tetrahedron. Bull. Amer. Math. Soc. 64 (1958) 90-91. 11. A connected subset of the plane. Fund. Math. 46 (1958) 15-24. 12. Arcwise connected sets in the plane. Duke Math. J. 30 (1963) 363-366. 360 CHAPTER 9

13. A technique for constructing examples. Proc. Amer. Math. Soc. 16 (1965) 1320-1323. 14. A new proof that metric spaces are paracompact. Proc. Amer. Math. Soc. 20 (1969) 603. 15. A note on certain function spaces. Arch. Math. 7 (1957) 469- 470. 16. Composants and N. Proc. Wash. State Univ. Conf. on General Topology (Pullman, Wash., 1970) Pi Mu Epsilon, Department of Mathematics, Washington State University, Pullman, Wash- ington, 1970. 17. Souslin’s conjecture. Amer. Math. Monthly 76 (1969) 113-119. 18. A normal space X for which X I is not normal. Bull. Amer. Math. Soc. 77 (1971) 246. × 19. Partial orders on the types in N. Trans. Amer. Math. Soc. 155 (1971) 353-362.

Tamano, Hisahiro* ranklin D. 02 • 1. New results on the normal Moore space problem. Proc. Wash- ington State Univer. Conf. on General Topology (Pullman, Wash., 1970) Pi Mu Epsilon, Department of Mathematics, Wash- ington State Univ. Pullman, Washington, 1970. ACADEMIC DESCENDANTS AND PUBLICATIONS 361 C. E. Burgess and his Mathematical Descendandts

01 Burgess, C. E. 02 Wiser, Horace C., University of Utah, 1961 02 Loveland, Lowell Duane, University of Utah, 1965 02 Cannon, Lawrence Orson, University of Utah, 1965 02 Lister, rederick M., University of Utah, 1965 02 Sher, Richard Benjamin, University of Utah, 1966 02 Lambert, Howard Wilson, University of Utah, 1966 02 Eaton, William Thomas, University of Utah, 1967 02 Lamoreaux, Jack Wayne, University of Utah, 1967 02 Pettey, Dix Hayes, University of Utah, 1968 02 Cannon, James Weldon, Oniversity of Oteh, 1969

Publications of R. D. Anderson and his Mathematical Descendants

Burgess, C. E. 01 • 1. Continua and their complementary domains in the plane. Duke Math. J. 18 (1951) 901-917. 2. Continua and their complementary domains in the plane. II. Duke Math. J. 19 (1952) 223-230. 3. Continua which are the sum of a finite number of indecomposable continua. Proc. Amer. Math. Soc. 4 (1953) 234-239. 4. Some theorems on n-homogeneous continua. Proc. Amer. Math. Soc. 5 (1954) 136-143. 5. Collections and sequences of continua in the plane. Pacific J. Math. 5 (1955) 325-333. 6. Certain types of homogeneous continua. Proc. Amer. Math. Soc. 6 (1955) 348-350. 7. Separation properties of n-indecomposable continua. Duke Math. J. 23 (1956) 595-599. 8. A note on the separation of connected sets by finite sets. Proc. Amer. Math. Soc. 7 (1956) 1115-1116. 362 CHAPTER 9

9. Continua and various types of homogeneity. Trans. Amer. Math. Soc. 88 (1957) 366-374. 10. Chainable continua and indecomposability. Paciﬁc J. Math. 9 (1959) 653-659. 11. Some conditions under which a homogeneous continuum is a simple closed curve. Proc. Amer. Math. Soc. 10 (1959) 613- 615. 12. Homogeneous continua which are almost chainable. Canadian J. Math. 13 (1961) 519-528. 13. Collections and sequences of continua in the plane. II. Paciﬁc J. Math. 11 (1961) 447-454. 14. Continua which have width zero. Proc. Amer. Math. Soc. 13 (1962) 477-481. 15. Properties of certain types of wild surfaces in E3. Amer. J. Math. 86 (1964) 325-338. 16. Characterizations f tame surfaces in E3. Trans. Amer. Math. Soc. 114 (1965) 80-07. 17. Pairs of 3-cells with intersecting boundaries in E3. Amer. J. Math. 88 (1966) 309-313. 18. Criteria for a 2-sphere in S3 to be tame modulo two points. Mich. Math. J. 14 (1967) 321-330. 19. A characterization of homogeneous plane continua that are circularly chainable. Bull. Amer. Math. Soc. 75 (1969 1354-1356. 20. Tame subsets of spheres in E3. Proc. Amer. Math. Soc. 22 (1969) 395-401.(with Cannon, J. W.) 21. Sequentially l-ULC surfaces in E3. Proc. Amer. Math. Soc. 1 (1968) 653-659.(with Loveland, L. D.)

Wiser, Horace Clark 02 • 1. Decomposition and homogeneity of continua on a 2-manifold. Paciﬁc J. Math. 12 (1962) 1145-1162. 2. Near-homogeneity on 2-manifolds. Amer. Math. Monthly 74 (1967) 423-426. 3. A note on polar topologies. Canad. Math. Bull. 11 (1968) 607-609.(with Robertson, J. M.) 4. Topologies of uniform convergence. Prace Mat. 12 (1969) 231- 235. ACADEMIC DESCENDANTS AND PUBLICATIONS 363

Loveland L. D. 02 • 1. Tame surfaces and tame subsets of spheres in E3. Trans. Amer. Math. Soc. 123 (1966) 355-368. 2. Wild points of cellular subsets of 2-spheres in S3. Mich. Math. J. 14 (1967) 427-431. 3. Conditions implying that a 2-sphere is almost tame. Trans. Amer. Math. Soc. 131 (1968) 170-181. 4. Tame subsets of spheres in E3. Paciﬁc J. Math. 19 (1966) 489- 517. 5. Suﬃcient conditions for a closed set to lie on the boundary of a 3-cell. Proc. Amer. Math. Soc. 19 (1968) 649-652. 6. Wild points of cellular subsets of spheres in S3. II. Mich. Math. J. 15 (1968) 191-192. 7. Cross sectionally continuous spheres in E3. Bull. Amer. Math. Soc. 75 (1969) 396-397. 8. Piercing locally spherical spheres with tame arcs. Illinois J. Math. 13 (1969) 327-330. 9. Piercing points of crumpled cubes. Trans. Amer. Math. Soc. 143 (1969) 145-152. 10. Tameness implied by extending a homeomorphism to a point. Proc. Amer. Math. Soc. 23 (1969) 287-293. 11. Sequentially l-ULC srfaces in E3. Proc. Amer. Math. Soc. 19 (1968) 653-659.(with Burgess, C. E.) 12. Surfaces of vertical ordr 3 are tame. Bull. Amer. Math. Soc. 23 (1969) 287-293.(with Jensen, R. A.)

Cannon, L. O. 02 • 1. Sums of solid horned spheres. Trans. Amer. Math. Soc. 122 (1966) 203-228. 2. On a point-segment decomposition of E3 deﬁned by McAuley. Proc. Amer. Math. Soc. 19 (1968) 624-630.

Lister, F. M. 02 • 1. Simplifying intersections ofdisks in Bing’s side approximation theorem. Paciﬁc J. Math. 22 (1967) 281-295. 364 CHAPTER 9

Sher, R. B. 02 • 1. Toroidal decompositions of E3. Fund. Math. 61 (1967/68) 225- 241. 2. A note on an example Of Stallings. Proc. Amer. Math. Soc. 19 (1968) 619-620. 3. Defining subsets of E3 by cubes. Pacific J. Math. 25 (1968) 613619. 4. Concerning wild Cantor sets in E3. Proc. Amer. Math. Soc. 19 (1968) 1195-1200. 5. Families of arcs in E3. Trans. Amer. Math. Soc. 143 (1969) 109-116. 6. A result on unions of flat cells. Duke Math. J. 37 (1970) 85-88. 7. A note on 0-dimensional decompositions of E3. Amer. Math. Monthly 75 (1968) 377-378.(with Alford, W. R.) 8. Defining sequences for compact 0-dimensional decompositions of E. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 17 (1969) 209-212. 9. Point-like 0-dimensional decompositions of S3. Pacific J. Math. 24 (1968 511-518.(with Lambert H. W.)

Lambert, H. W. 02 • 1. A topological property to Bing’s decomposition of E3 into points and tame arcs. Duke Math. J. 34 (1967) 501-509. 2. Toroidal decompositions of E3 which yield E3. Fund. Math. 61 (1967) 121-132. 3. Mapping cubes with holes onto cubes with handles. Illinois J. Math. 13 (1969) 606-615. 4. Point-like 0-dimensional decompositions of S3. Paciﬁc J. Math. 24 (1968) 511-518.(with Sher, R. B.)

Eaton, William T. 02 • 1. Side approximations in crumpled cubes. Duke Math. J. 35 (1968) 707-719. 2. Cross sectionally simple spheres. Bull. Amer. Math. Soc. 75 (1969) 375-378. ACADEMIC DESCENDANTS AND PUBLICATIONS 365

3. A note about locally spherical spheres. Canadian J. Math. 21 (1969) 1001-1003. 4. Taming a surface by piercing with disks. Proc. Amer. Math. Soc. 22 (1969) 724-727. ; 5. A dense set of sewings of two crumpled cubes yields S3. Fund. Math. 65 (1969) 51-60.(with Daverman, Robert J.) 6. An equivalence for the embeddings of cells in a 3-manifold. Trans. Amer. Math. Soc. 145 (1969) 369-381.

Lamoreaux, Jack W. 02 • 1. Decomposition of metric spaces with a 0-dimensional set of nondegenerate elements. Canadian J. Math. 21 (1969) 202-216.

Cannon, James Weldon 02 • 1. An elementary proof of the Jordan-Holder theorem. Amer. Math. Monthly 75 (1968) 279. 2. Tame subsets of spheres in E3. Proc. Amer. Math. Soc. 22 (1969) 395-401.(with Burgess, C. E.) 3. Embeddings of surfaces in E3. Rocky Mountain Journal Math. (1971) no. 2, 259-344. 4. Sets which can be missed by side approximations to spheres. Paciﬁc J. Math. 34 (1970) 321-334. 5. and Wayment, S. G. 6. An embedding problem. Proc. Amer. Math. Soc. 25 (1970) 566-570. 7. Characterization of taming sets on 2-spheres. Trans. Amer. Math. Soc. 147 (1970) 289-299.

Burgess, C. E. 01 • 1. and Cannon, J. W. 2. Embeddings of surfaces in E3. Rocky Mountain J. Math. (1971) no. 2, 259-344.

Loveland, L. D. 02 • 1. A 2-sphere of vertical order 5 bounds a 3-cell. Proc. Amer. Math. Soc. 26 (1970) 674-678. 366 CHAPTER 9

2. A surface in E3 is tame if it has round tangent balls. Trans. Amer. Math. Soc. 152 (1970) 388-397.

Lister F. M. 02 • 1. Tame boundary sets of crumpled cubes in E3. Proc. Amer. Math. Soc. 25 (1970) 377-378.

Sher, R. B. 04 • 1. Deﬁning subsets of manifolds by cells. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phy. 17 (1969) 363-365. 2. Geometries embedding invariants of simple closed curves in the threespace. Duke Math. J. 36 (L969) 683-693. 3. Determining the cellularity of a i-complex by properties of its arcs. Proc. Amer. Math. Soc. 26 (1970) 491-498. 4. Tame polyhedra in wild cells and spheres. Proc. Amer. Math. Soc. 30 (1971) 169-174.

Lambert H. W. 02 • 1. Replacing certain maps of 3-manifolds by homeomorphisms. Proc. Amer. Math. Soc. 23 (1969) 676-678. 2. A l-linked link whose longitudes be in the second commutator subgrup Trans. Amer. Math. Soc. 147 (1970) 261-269.

Pettey, D. H. 02 • 1. One-to-one mappings into the plane. Fund. Math. 67 (1970) 209-218. 2. Mappings onto the plane. Trans. Amer. Math. Soc. 157 (1971) 297-309. ACADEMIC DESCENDANTS AND PUBLICATIONS 367 B. J. Ball and his Mathematical Descendandts

01 Ball, B. J. 02 May, J. Gaylord, University of Virginia, 1960 02 May, W. Graham, University of Virginia, 1960 02 Ford , Jo W., Auburn University , 1964 03 Beck, Oscar, Auburn University, 1971 02 Gentry, Karl R., University of Georgia, 1965 02 Hodge , James E ., University of Georgia , 1965 02 Pittman, Chatty R. , University of Georgia, 1965 02 Transue, William R. R., University of Georgia, 1967 02 Connor, Andrew C. , University of Georgia, 1969

Publications of B. J. Ball and his Mathematical Descendants

Ball, B. J. 01 • 1. Continuous and equicontinuous collections of arcs. Duke Math. J. 19 (1952) 423-433. 2. Some theorems concerning spirals in the plane. Amer. J. Math. 76 (1954) 66-80. 3. Countable paracompactness in linearly ordered spaces. Proc. Amer. Math. Soc. 5 (1954) 190-192. 4. A note on the separability of an ordered space. Canadian J. Math. 7 (1955) 548-551. 5. The normality of the product of two ordered spaces. Duke Math. J. 24 (1957) 15-18. 6. The sum of two solid horned spheres. Ann. of Math. (2) 69 (1959) 253-257. 7. Certain collections of arcs in E3. Proc. Amer. Math. Soc. 10 (1959) 699-705. 8. Penetration indices and applications. Topology of 3-manifolds and related topics. (Proc. the Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliﬀs, N. J., 1962, 37-39. 368 CHAPTER 9

9. Finite collections of 2-spheres in E3. Proc. Amer. Math. Soc. 13 (1962) 1-8. 10. Some almost polyhedral wild arcs. Duke Math. J. 30 (1963) 33-38. (with Alford, W. R.) 11. Extending real maps deﬁned on a subset of a disk. Proc. Amer. Math. Soc. 20 (1969) 75-80.(with Ford, Jo; Thomas, E. S., Jr.)

Ford, Jo 02 • 1. Imbedding collections of compact O-dimensional subsets of E2 in continuous collections of mutually exclusive arcs. Fund. Math. 58 (1966) 45-57. 2. Extending real aps deﬁned on a subset of a disk. Proc. Amer. Math. Soc. 20 (1969) 75-80.(with Ball, B. J.; Thomas, E. S., Jr.) 3. Aligning functions deﬁned on Cantor sets. Trans. Amer. Math. Soc. 141 (1969) 63-69.(with Thomas, E. S., Jr.)

Gentry, K. R. 02 • 1. Some properties of the induced map. Fund. Math. 66 (1969/70) 55-59.

Transue, W. R. R. 02 • 1. The local characterization of vector function spaces with duals of integral type. J. Analyse Math. 6 (1958) 225-260. 2. On the hyperspace of subcontinua of the pseudoarc. Proc. Amer. Math. Soc. 18 (1967) 1074-1075. 3. On embedding cones over circularly chainable continua. Proc. er. Math. Soc. 21 (1959) 275-276.(with Bennett, Ralph) 4. The existence of vector function spaces with duals of integral type. Colloq. Math. 6 (1958) 95-117.(with Morse, Marston)

Gentry, K. R. 02 • 1. , Hoyle, H. B. 2. Somewhat continuous functions. Czech. Math. J. 21 (96) (1971) 5-12. 3. 1 ACADEMIC DESCENDANTS AND PUBLICATIONS 369

Pittman, C. R. 02 • 1. An elementary proof of the triod theorem. Proc. Amer. Math. Soc. 25 (1970) 919.

Transue, W. R. R. 02 • 1. and Hinrichsen, J. W.; Fitzpatrick, B. 2. Concerning upper semicontinuous decompositions of irreducible contlnua. Proc. Amer. Math. Soc. 30 (1971) 157-163. 370 CHAPTER 9 Elden Dyer and his Mathematical Descendandts

01 Dyer, Elden 02 Gershenson, Hillel, University of Chicago, 1961 02 Kirby, Robion, University of Chicago, 1964 03 Turner, Edward, U. C. L. A., 1968 03 Gauld,02 McNamara, David, U. C. James,L. A., 1969 University of Chicago, 1964 02 Miller, John, Rice University, 1967 02 Braun, J., City University of New York, 1971

Publications of Elden Dyer and his Mathematical Descendants

Dyer, Elden 01 • 1. Irreducibility of the sum of the elements of a continuous collection of continua. Duke Math. J. 20 (1953) 589-592. 2. Continuous collections of decomposable continua on a spherical surface. Proc. Amer. Math. Soc. 6 (1955) 351-360. 3. Certain transformations which lower dimension. Ann. of Math. (2) 63 (1956) 15-19. 4. A fixed point theorem. Proc. Amer. Math. Soc. 7 (1956) 662-672. 5. Regular mappings and dimension. Ann. of Math. (2) 67 (1958) 119-149. 6. On the dimension of products. Fund. Math. 47 (1959) 141-160. 7. Chern characters of certain complexes. Math Zeit. 80 (1962/63) 363-373. 8. The functors of algebraic topology. Studies in Modern Topology, Math. Assoc. (distributed by Prentice-Hall, Englewood Cliffs, N. J.) 1968, 134-164. 9. Some spectral sequences associated with fibrations. Trans. Amer. Math. Soc. 145 (1969) 397-437.(with Kahn, Daniel S.) 10. On singular fiberings by spheres. Mich. Math. J. 6 (1959) 303- 311.(with Conner, P. E.) ACADEMIC DESCENDANTS AND PUBLICATIONS 371

11. Completely regular mappings. Fund. Math. 45 (1958) 103-118. (with Hamstrom, M. E.) 12. Regular mappings and the space of homeomorphisms on a 2- manifold. Duke Math. J. 25 (1958) 521-531. 13. Homology of principal bundles. Proc. Sympos. Pure Math., A.M.S., Providence, R. I., 3 (1961) 101-108.(with Lashof, R. K.) 14. A topological proof of the Bott periodicity theorems. Ann. Math. Pure Appl. (4) 54 (1961) 231-254. 15. Homology of iterated loop spaces. Amer. J. Math. 84 (1962) 35-38. 16. Connectivity of topological lattices. Paciﬁc J. Math. 9 (1959) 443-448.(with Shields, Allen)

Gershenson, H. H. 02 • 1. Relationships between the Adams spectral sequence and Toda’s calculations of the stable homotopy groups of spheres. Math Zeit. 81 (1963) 223-259. 2. Higher composition products. J. Math. Kyoto Univ. 5 (1965) 1-37. 3. On the homotopy groups of suspensions of complex Grassmann manifolds. Topology 8 (1969) 59-67.

Kirby, Robion 02 • 1. Smoothing locally flat imbeddings. Bull. Amer. Math. Soc. 72 (1966) 147-148. 2. On the annulus conjecture. Proc. Amer. Math. Soc. 17 (1966) 178185. 3. The union of flat (n balls is flat in Rn. Bull. Amer. Math. Soc. 74 (1968) 614-617. . 4. On the set of non-locally flat points of a submanifold of codimension one. Ann. of Math. (2) 88 (1968) 281-290. 5. Stable homeomorphisms and the annulus conjecture. Ann. of Math. (2) 89 (1969) 575-582. 6. On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75 (1969) 742-749.(with Siebenmann, L. C.) 372 CHAPTER 9

Turner, Edward 03 • 1. Diﬀeomorphisms of a product of spheres. Invent. Math. 8 (1969) 69-82.

Miller, John 02 • 1. On the resolvent of a linear operation associated with a well- posed Cauchy problem. Math. Comp. 22 (1968) 541-548.

Dyer, Eldon 01 • 1. Cohomology theories. Mathematics Lecture Note Series. W. W. Benjamin, Inc., New York-Amsterdam, 1969 xiii+183 pp.

Kirby, R. C. 02 • 1. Co-dimension-two locally ﬂat imbeddings have normal bundles. Topology of Manifolds. (Proc. Conf. Univ. Georgia, Athens, Georgia, 1969) 416-423. Markham, Chicago, Illinois, 1970.

Gauld, David 03 • 1. Mersions of topological manifolds. Trans. Amer. Math. Soc. 149 (1970) 539-560.

Miller, John 02 • 1. A diﬀerence equation satisﬁed by the functions u of Harish Chan- dra. Amer. J. Math. 92 (1970) 362-368.(with Simms, D. J.)

Hamstrom, Mary Elizabeth 01 • 1. Linear independence in Abelian groups. Proc. Amer. Math. Soc. 2 (1951) 487-489. 2. Concerning continuous collections of continuous curves. Proc. Amer. Math. Soc. 4 (1953) 240-243. 3. Concerning certain types of webs. Proc. Amer. Math. Soc. 4 (1953) 974-978. 4. Concerning webs in the plane. Trans. Amer. Mat. Soc. 74 (1953) 500-513. 5. Concerning the imbedding of upper semicontinuous collections of continua in continuous collections of continua. Amer. J. Math. 76 (1954) 793-810. ACADEMIC DESCENDANTS AND PUBLICATIONS 373

6. A characterization of plane continuous curves which are filled up by continuous collections of continuous curves. Amer. J. Math. 77 (1955) 914-928. 7. Regular mappings whose inverse are 3-cells. Amer. J. Math. 82 (1960) 393-429. 8. Regular mappings and the space of homeomorphisms on a 3- manifold. Mem. Amer. Math. Soc. No. 40 (1961) 42 pp. 9. Some global properties of the space of homeomorphisms on a disc with holes. Duke Math. J. 29 (1962) 657-662. 10. A decomposition theorem for E4. Illinois J. Math. 7 (1963) 503- j 507. 11. The space of homeomorphisms on a torus. Illinois J. Math. 9 (1965) 59-65. 12. Homotopy properties of the space of homeomorphisms on p2 and the Klein bottle. Trans. Amer. Math. Soc. 120 (1965) 37-45. 13. Homotopy groups of the space of homeomorphisms on a 2-manifold. Illinois J. Math. 10 (1966) 563-573. 14. A note on continuous collections of continuous curves filling up a continuous curve in a plane. Proc. Amer. Math. Soc. 5 (1954)(with Anderson, R. D.) 748-752. 15. On spaces filled up by continuous collections of atroidic continuous curves. Proc. Amer. Math. Soc. 6 (1955) 766-769. 16. and R. P. 17. Collapsing a triangulation of a ”knotted” cell. Proc. Amer. Math. Soc. 21 (1969) 327-331. 18. Regular mappings: A survey. Proc. First Conf. on Mono- tone Mappings and Open Mappings (SUNY at Binghamton, New York, 1970) 238-254. State University of New York at Bingham- ton, 3inghamton, N. Y., 1971. 19. Completely regular mappings whose inverses have LC homeomorphism group: A correction. Proc. First Conf. on Mono- tone Mappings and Open Mappings (SUNY at Binghamton, New York, 1970) 255-260. State University of New York at Bingham- ton, Binghamton, N. Y., 1971.

Slye, John M. 01 • 1. Flat spaces for which the Jordan curve theorem holds true. Duke Math. J. 22 (1955) 143-151. 374 CHAPTER 9

2. Collections whose sums are two-manifolds. Duke Math. J. 24 (1957) 275-298.

Mohat, J. T. 01 • 1. Concerning spirals in the plane. Duke Math. J. 24 (1957) 249264. ACADEMIC DESCENDANTS AND PUBLICATIONS 375 B. J. Pearson and his Mathematical Descendandt

01 Pearson, B. J. 02 Miller, Gary Glenn, University of Missouri, Kansas City, 1968

Publications of B. J. Pearson and his Mathematical Descendant

Pearson, B. J. 01 • 1. A connected point set, in the plane which spirals down on each of its points. Duke Math. J. 25 (1958) 603-613.

Miller, G. G. 02 • 1. Countable connected spaces. Proc. Amer. Math. Soc. 26 (1970) 355-360. 376 CHAPTER 9 Steve Armentrout and his Mathematical Descendandts

01 Armentrout, Steve 02 Martin, Joseph M., University of Iowa, 1962 03 Rolfsen, Dale P., University of Wisconsin, 1967 03 White, Warren, University of Wisconsin, 1967 03 Mason, William, University of Wisconsin, 1968 03 Simon, Jonothan, University of Wisconsin, 1969 03 Moser, Louise, University of Wisconsin, 1970 03 Mayland,02 Meyer, James, Donald University V., of University Wisconsin, of 1971 Iowa, 1962 02 Alzoobaee, Arabi H., University of Iowa, 1962 02 Lininger, Lloyd L., University of Iowa, 1964 02 Fugate, J. Brauch, University of Iowa, 1964 03 Bennett,02 Schori, Donald Richard Earl, University M., University of Kentucky, of Iowa, 1970 1964 02 Anderson, Bruce A., University of Iowa, 1966 03 Richardson,02 Hutchinson, Joan, Arizona Thomas State C., University, University 1969 of Iowa, 1966 02 Voxman, William L., University of Iowa, 1968 02 Neuzil, John P., University of Iowa, 1969 02 Summerhill, R. Richard, University of Iowa, 1969 ACADEMIC DESCENDANTS AND PUBLICATIONS 377 Publications of Steve Armentrout and his Mathematical Descendants

Armentrout, Steve 01 • 1. Concerning a certain collection of spirals in the plane. Duke Math. J. 26 (1959) 243-250. 2. Separation theorems for some plane-like spaces. Trans. Amer. Math. Soc. 97 (1960) 120-130. 3. I A Moore space on which every real-valued continuous function is constant. Proc. Amer. Math. Soc. 12 (1961) 106-109. 4. Upper semi-continuous decompositions of E3 with at most countably many non-degenerate elements. Ann. of Math. (2) 78 (1963) 605-618. 5. Decompositions of E3 with a compact 0-dimension set of nondegenerate elements. Trans. Amer. Math. Soc. 123 (1966) 165-177. 6. Concerning a wild 3-cell described by Bing. Duke Math. J. 33 (1966) 689-704. 7. Concerning cellular decompositions of 3-manifolds that yield 3- manifolds. Trans. Amer. Math. Soc. 133 (1968) 307-332. 8. Concerning cellular decompositions of 3-manifolds with boundary. Trans. Amer. Math. Soc. 137 (1969) 231-236. 9. Cellular decompositions of 3-manifolds that yield 3-manifolds. Bull. Amer. Math. Soc. 75 (1969) 453-456. 10. Completing Moore spaces. Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968 22-35. 11. Shrinkability of certain decompositions of E3 that yield E3. Illi- nois J. Math. 13 (1969) 700-706. 12. A toroidal decomposition of E3. Fund. Math. 60 (1967) 81-87. (with Bing, R. H.) 13. Monotone decompositions of E3. Topology Seminar (Wisconsin, 1965) Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N. J., 1966, 1-25. 14. On embedding decomposition spaces of En in En 1. Fund. Math. 61 (1967) 1-21. 378 CHAPTER 9

15. Equivalent decomposition of R3. Paciﬁc J. Math. 24 (1968) 205-227.(with Lininger, L. L.: Meyer, D. V.) 16. Decompositions into compact sets with W properties. Trans. Amer. Math. Soc. 141 (1969) 433-442.(with Price, Thomas M.)

Martin, Joseph M. 02 • 1. Homogeneous countable connected Hausdorﬀ spaces. Proc. Amer. Math. Soc. 12 (1961) 308-314. 2. A note on uncountably many disks. Paciﬁc J. Math. 13 (1963) 13311333. 3. Extending a disk to a sphere. Trans. Amer. Math. Soc. 109 (1963) 385-399. 4. Tame arcs on disks. Proc. Amer. Math. Soc. 16 (1965) 131-133. 5. A rigid sphere. Fund. Math. 59 (1966) 117-121. 6. The impossibility of desuspending coliapses. Bull. Aer. Math. Soc. 74 (1968) 979-981.(with Lin, S. Y. . B. R.) 7. Triangulations ohthe 3-ball with knotted spanning l-simplexes and collapsible R derived subdivisions. Trans. Amer. Math. Soc. 137 (1969) 451-458. 8. Homotopic arcs are isotopic. Proc. Amer. Math. Soc. 19 (1968) 1290-192.(with Rolfsen, Dale)

Rolfsen Dale P. 03 • 1. . 2. Alternative metrization proofs. Canadian J. Math. 18 (1966) 750 757. 3. Strongly convex metrics in cells. Bull. Amer. Math. Soc. 74 (l968) 171-175. 4. Homotopicarcs are isotopic. Proc. Amer. Math. Soc. 19 (1968) 1290-1292.(with Martin, Joseph)

White, Warren 03 • 1. Some tameness conditions involving singular disks. Trans. Amer. Math. Soc. 143 (1969) 223-234.

Mason William 03 • ACADEMIC DESCENDANTS AND PUBLICATIONS 379

1. . 2. Homeomorphic continuous curves in 2-space are isotopic in 3- space. Trans. Amer. Math. Soc. 142 (1969) 700-706.

Meyer, Donald V. 03 • 1. A decomposition of E3 into points and a null family of tame 3-cells in E3. Ann. of Math. (2) 78 (1963) 600-604. 2. E3 modulo a 3-cell. Paciﬁc J. Math. 13 (1963) 193-196.

Alzoobaee, Orabi H. 02 • 1. Product of countably many complete Moore spaces. Bull. Col- lege Sci. (Baghdad) 9 (1966) 197-208. 2. Normal Moore spaces. Bull. College Sci. (Baghdad) 9 (1966) 201-208.

Fugate, J. B. 02 • 1. Decomposable chainable continua. Trans. Amer. Math. Soc. 123 (1966) 460-468. 1 2. A characterization of chainable continua. Canadian J. Math. 21 (1969) 383-393.

Schori, R. M. 02 • 1. A universal snake-like continuum. Proc. Amer. Math. Soc. 16 (1965) 1313-1316. 2. Inverse limits and homogeneity. Trans. Amer. Math. Soc. 124 (1966) 533-539. 3. Hyperspaces and symmetric products of topological spaces. Fund. Math. 63 (1968) 77-88. 4. A factor theorem for Frechet manifolds. Bull. Amer. Math. Soc. 75 (1969) 53-56.(with Anderson, R. D.) 5. Factors of infinite-dimensional manifolds. Trans. Amer. Math. Soc. 142 (1969) 315-330. 6. Topological classification of infinite dimensional manifolds by homotopy type. Bull. Amer. Math. Soc. 76 (1970) 121-124.(with Henderson, David W.)

Anderson B. A. 02 • 380 CHAPTER 9

1. Chains of metric topologies. Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968, 2-14. 2. Tl-complements of Tl topologies. Proc. Amer. Math. Soc. 23 (1969) 77-81.(with Stewart, D. G.)

Armentrout, Steve 01 • 1. A decomposition of E3 onto straight arcs and singletons. Dis- sertationes Math. Rozprawy Mat. 68 (1970) 46 pp. 2. A three dimensional spheroidal space which is not a sphere. und. Math. 68 (1970) 183-186. 3. W properties of compact sets. Trans. Amer. Math. Soc. 143 (1969) 487-498. 4. Homotopy properties of decomposition spaces. Trans. Amer. Math. Soc. 143 (1969) 499-507. 5. On the strong local simple connectivity of the decomposition spaces of toroidal decompositions. Fund. Math. 69 (1970) 15- 37. 6. Local properties of decomposition spaces. Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Bing- hamton, Binghamton, N.Y., 1970) State Univ. of New York at Binghamton, Binghamton, N. Y., 1971.

Martin, J. M. 02 • 1. and Bing, R. H. 2. Cubes with knotted holes. Trans. Amer. Math. Soc. 155 (1971) 217-231. 3. Monotone images of E3. Proc. First Conf. on Monotone Map- pings and Open Mappings (SUNY at Binghamton, Binghamton, N. Y., 1970) State University of New York at Binghamton, Bing- hamton, N. Y., 1971.

Rolfsen, D. P. 03 • 1. Characterizing the 3-cell by its metric. Fund. Math. 68 (1970) 215-223.

White, Warren 03 • ACADEMIC DESCENDANTS AND PUBLICATIONS 381

1. A 2-complex is collapsible if and only if it admits a strongly convex metric. Fund. Math. 68 (1970) 23-29. Meyer, D. V. 02 • 1. More decompositions of E which are factors of En . Fund. Math. 7 (1970) 49-55. Lininger, L. L. 02 • 1. Decompositions of a 3-cell. Fund. Math. 68 (1970) 1-6. 2. Actions on S . Topology 9 (1970) 301-308. 3. and Goldstein, Richard Z. 4. Stability in spin structures. Topology of Manifolds. (Proc. Inst. Univ. Georgia, Athens, Georgia, 1969) Markham, Chicago, Illi- nois, 1970. 268-273. 5. Sl actions on 6-manifolds. Topology 9 (1971) 301-308. Anderson, B. A. 02 • 1. A class of topologies with Tl-complements. Fund. Math. 69 (1970) 267-277. 2. Families of mutually complementary topologies. Proc. Amer. Math. Soc. 29 (1971) 362-368. 3. -427 Voxman, W. L. 02 • 1. On the shrinkability of decompositions of 3-manifolds. Trans. Amer. Math. Soc. 150 (1970) 27-39. 2. Nondegenerately continuous decompositions of 3-manifolds. Fund. Math. 68 (1970) 307-320. 3. On the uniform of certain cellular decompositions of 3-manifolds. Illinois J. Math. 15 (1971) 387-392. Neuzil, J. P. 02 • 1. Spheroidal decompositions of E4. Trans. Amer. Math. Soc. 155 (1971) 35-64. Summerhill, Richard R. 02 • 1. Tree-like continua and cellularity. Proc. Amer. Math. Soc. 26 (1970) 201-205. 382 CHAPTER 9 W. S. Mahavier and his Mathematical Descendandts

Mahavier, W. S. 02 Bacon, Philip, University of Tennessee, 1964 02 Bennett, Ralph B., University of Tennessee, 1964 03 Williams, Jerry F., Auburn University, 1970 03 Van Cleave, John T., Auburn University, 1971 02 Russell, Mary Jean, Emory University, 1969 ACADEMIC DESCENDANTS AND PUBLICATIONS 383 Publications of W. S. Mahavier and his Mathematical Descendants

Mahavier, W. S. 01 • 1. Rates of change and functional relations. Fund. Math. XLVIII (1960) 265-269. 2. A chainable continuum not homomorphic to an inverse limit on (0,1) with only one bonding map. Proc. Amer. Math. Soc. 18 (2) (1967) 284-285. 3. Upper semi-continuous decompositions of irreducible continua. Fund. Math. 60 (1967) 53-57. 4. Semi-groups on chainable and circle-like continua. Math. Zeit. 106 (1968) 159-161.(with Friedberg, Michael) 5. Continua with only two diﬀerent subcontinua. Proc. Topology Conference, Ariz. State Univ., Tempe, Arizona, 1967. 203-206. 6. Arcs in inverse limits on (0,1) with only one -onding map. Proc. Amer. Math. Soc. (3) 21 (1969) 587-590 7. Atomic mappings on irreducible Hausdorﬀ continua. Fund. Math. Bacon, Philip 02 • 1. Coincidences of real-valued maps from the n-torus. Fund. Math. 57 (1965) 63-89. 2. Equivalent formulations of the Borsuk-Ulam theorem. Canad. J. Math. 18 (1966) 492-502. 3. Extending a complete metric. Amer. Math. Monthly 75 (1968) 642643.

Bennett, Ralph B. 02 • 1. Embedding products of chainable continua. Proc. Amer. Math. Soc. 16 (1965) 1026-1027. 2. Locally connected 2-cell and 2-sphere-like continua. Proc. Amer. Math. Soc. 17 (1966) 674-681. 3. On selfproductive sets. Colloq. Math. 17 (1967) 309-313. 4. On embedding cones over circularly chainable disks. Proc. Amer. Math. Soc. 21 (1969) 275-276.(with Transue, W. R. R.)

Bacon, Philip 02 • 384 CHAPTER 9

1. The compactness of countably compact spaces. Paciﬁc J. Math. 32 (1970) 587-592. 2. A characterization of locally connected unicoherent continua. J. Austral. Math. Soc. 10 (1969) 257-265. 3. An acyclic continuum that admits no mean. Fund. Math. 67 (1970) 11-13. 4. Unicoherence in means. Colloq. Math. 21 (1970) 211-215. ACADEMIC DESCENDANTS AND PUBLICATIONS 385 L. B. Treybig and his Mathematical Descendandts

02 Bell, Harold, Tulane University, 1964 02 Babcock, William Warren, Tulane University, 1965 02 Penney, David Emory, III, Tulane University, 1966 03 Cliett, Otis Jay, University of Georgia, 1970 03 Boren, Nancy Susan, University of Georgia, 1970 02 Riley, John Philip, Jr., Tulane University, 1968 02 Martin, John, Tulane University, 1970

Publications of L. B. Treybig and his Mathematical Descendants

Treybig, L. B. 01 • 1. Concerning local separability in locally peripherally separable spaces. Proc. Amer. Math. Soc. 10 (1959) 957-958. 2. Separability in metric spaces. Duke Math. J. 27 (1960) 383-386. 3. Concerning certain locally peripherally separable spaces. Paciﬁc J. Math. 10 (1960) 697-704. 4. Concerning local separability in metric spaces. Duke Math. J. 29 (1962) 125-128. 5. Concerning homogeneity in totally ordered, connected topological space. Paciﬁc J. Math. 13 (1963) 1417-1421. 6. Concerning continua which are continuous images of compact ordered sets. Duke Math. J. 32 (1965) 417-422. 7. A characterization of the double point structure of the projection of polygonal knot in regular position. Trans. Amer. Math. Soc. 130 (1968) 223-247. 8. Prime mappings. Trans. Amer. Math. Soc. 130 (1968) 248-253. 9. Concerning upper semi-continuous decomposition of E whose nondegenerate elements are polyhedrals or star-like continua. Canad. J. Math. 20 (1968) 1308-1314.

Bell, Harold 02 • 386 CHAPTER 9

1. On ﬁxed point properties of plane continua.Trans. Amer. Math. Soc. 128 (1967) 539-548.

Penney, David E., III 02 • 1. Generalized Brunnian links. Duke Math. J. 36 (1969) 31-32.

Treybig, L. B. 01 • 1. An approach to the polygonal knot problem using projections and isotopies. Trans. Amer. Math. Soc. 158 (1971) 409-421. 2. Concerning a bound problem in knot theory. Trans. Amer. Math. Soc. 158 (1971) 423-436.

Riley, John Philip 02 • 1. Decompositions of E3 and the tameness of their sets of nondegenerate elements. Duke Math. J. 38 (1971) 255-258. ACADEMIC DESCENDANTS AND PUBLICATIONS 387 J. N. Younglove and his Mathematical Descendandt

01 Younglove, J. N. 02 Bandy, Carroll L., University of Houston, 1972

Publications of J. N. Younglove and his Mathematical Descendant

Younglove, J. N. 01 • 1. Concerning dense metric subspaces of certain non-metric spaces. Fund. Math. 48 (1959) 15-25. 2. A theorem on metrization of Moore spaces. Proc. Amer. Math. Soc. 12 (1961) 592-593. 3. Separability and compactness in pointwise paracompact spaces. Amer. Math. Monthly 71 (1964) 412-413. 4. Two conjectures in point set theory. Topology Seminar (Wiscon- sin, 1965) 121-123. Annals of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N. J., 1966. 5. On normality and pointwise paracompactness. Paciﬁc J. Math. 25 (1968) 193-196.(with Traylor, D. R.) 6. A locally connected, complete Moore space on which every real- valued continuous function is constant. Proc. Amer. Math. Soc. 20 (1969) 527-530. 388 CHAPTER 9 G. W. Henderson and his Mathematical Descendandt

01 Henderson, G. W. 02 Krajewski, Lawrence L., University of Wisconsin, Milwaukee, 1971

Publications of G. W. Henderson and his Mathematical Descendant

Henderson, G. W. 01 • 1. Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc. Annals of Math. 27 (1960) 421-428. 2. The pseudo-arc as an inverse limit with one bonding map. Duke Math. J. 31 (1964) 421-425. 3. The mod 2 homology of the image of an exactly 2 to 1 map from a sphere. Proc. Amer. Math. Soc. 18 (1967) 723-726. 4. On the hyperspace of sub-continua of an arc-like continuum. Proc. Amer. Math. Soc. 27 (1971) 416-417.

Krawjewski, Lawrence L. 02 • 1. On expanding locally ﬁnite collections. Canad. J. Math. 23 (1971) 56-68. ACADEMIC DESCENDANTS AND PUBLICATIONS 389 Publications of J. M. Worrell

Worrell, J. M., Jr. 01 • 1. Upper semi-continuous decompositions of developable spaces. Proc. Amer. Math. Soc. 16 (1965) 485-490. 2. A characterization of metacompact spaces. Portugal. Math. 25 (1966) 171-174. 3. The closed continuous images of metacompact topological spaces. Portugal. Math. 25 (1966) 175-179. 4. Upper semicontinuous decompositions of spaces having bases of countable order. Portugal. Math. 26 (1967) 493-504. 5. On continuous mappings of metacompact Cech complete spaces. Pacific J. Math. 30 (1969) 555-562. 6. On collections of domains inscribed in a covering of a space in the sense of Alexandorff and Urysohn. Portugal. Math. 26 (1967) 405-420. 7. Characterizations of developable topological spaces. Canad. J. Math. 17 (1965) 820-830.(with Wicke, H. H.) 8. Open continuous mappings of spaces having bases of countable order. Duke Math. J. 34 (1967) 255-271. 9. Errats to: Open continuous mappings of spaces having bases of countable order. Duke Math. J. 34 (1967) 813-814. 10. Upper semicontinuous decompositions of spaces having bases of countable order. Portugal Math. 26 (1967) 493-504. 11. On collections of domains inscribed in a covering of a space in the sense of Alexandroff and Urysohn. Portugal. Math. 26 (1967) 405-420. 12. Extension of a result of Dieudonne. Proc. Amer. Math. Soc. 25 (1970) 634-637.(with Wicke, H. H.) 390 CHAPTER 9 Howard Cook and his Mathematical Descendandts

01 Cook, Howard 02 Ingram, William T., III, Auburn University, 1964 03 Kuykendall, Daniel, University of Houston, 1971

Publications of Howard Cook and his Mathematical Descendants

Cook, Howard 01 • 1. Concerning connected and dense subsets of indecomposable continua. Fund. Math. 53 (1963) 21-23. 2. On subsets of indecomposable continua. Colloq. Math. 13 (1964) 37-43. 3. Solution to problem P100. Colloq. Math. 13 (1964) 125. 4. On the most general plane closed point set through which it is possible to pass a pseudo-arc. Fund. Math. 55 (1964) 11-22. 5. Continua which admit only the identity mapping onto non-degenerate subcontinua. Fund. Math. 60 (1967) 241-249. 6. Upper semi-continuous continuum-valued mappings onto circle- like continua. Fund. Math. 60 (1967) 233-239. 7. Concerning three questions of Burgess about homogeneous continua. Colloq. Math. 19 (1968) 241-244. 8. Inverse limits of perfectly normal spaces. Proc. Amer. Math. Soc. 19 (1968) 189-192.(with Fitzpatrick, Ben, Jr.) 9. A characterization of indecomposable compact continua. Proc. of the Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona(with Ingram, W. T.) State Univ., Tempe, Ariz., 1968, 168-169. 10. Tree-likeness of dendroids and -dendroids. Fund. Math. 68 (1970) 19-22.

Ingram, William T. 02 • ACADEMIC DESCENDANTS AND PUBLICATIONS 391

1. Concerning non-planar circle-like continua. Canad. J. Math. 19 (1967) 242-250. 2. Decomposable circle-like continua. Fund. Math. 63 (1968) 193198. 3. A characterization of indecomposable compact continua. Proc. of the Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968, 168-169.(with Cook, Howard) 4. Obtaining AR-like continua as inverse limits with only two bonding maps. Glasnik Mat. Ser. III 4 (24) (1969) 309-312.

Cook, Howard 01 • 1. Tree-likeness of dendroids and -dendroids. Fund. Math. 68 (1970 19-22. 2. Tree-likeness of hereditarily equivalent continua. Fund. Math. 68 (1970) 203-205. 392 CHAPTER 9 James L. Cornette and his Mathematical Descendandts

01 Cornette, James L. 02 Chewning, William Carroll, Jr., University of Virginia, 1970 02 Girolo, Jack E., Iowa State University, 1971

Publications of James L. Cornette and his Mathematical Descendants

Cornette, J. L. 01 • 1. Continuum accessibility of points in totally disconnected sets. Duke Math. J. 32 (1965) 103-113. 2. Connectivity functions and images on Peano continuum. Fund. Math. 58 (1966) 183-192 3. Connectivity retracts of ﬁnitely coherent Peano continua. Fund. Math. 61 (1967) 177-182.(with Girolo, J. E.) 4. Retracts of the pseudo arc. Colloq. Math. 19 (1968) 235-239.

Girolo, Jack E. 02 • 1. Connectivity retracts of ﬁnitely coherent Peano continua. Fund. Math. 61 (1967) 177-182.(with Cornette, J. L.) ACADEMIC DESCENDANTS AND PUBLICATIONS 393 Publications of Jack W. Rogers

Rogers, Jack W. 01 • 1. A space whose regions are the simple domains of another space. Seria I. Prace Mathematyczne XIII (1970) 141-159. 2. An arc as the inverse limit of a single nowhere strictly monotonic bonding map on (0, 1). Proc. Amer. Math. Soc. 19, No. 13 (1968) 634-638. 3. Decomposable inverse limits with a single bonding map on (0,1) below the identity. Fund. Math. 66 (1970) 177-183. 4. Continua that contain only degenerate continuous images of plane continuum. Duke Math. J. 38 No. 3 (1970) 479-483. 5. Continua not an inverse limit with a single bonding map on a polyhedron. Proc. Amer. Math. Soc. 21 (1969) 281-283. 6. On universal tree-like continua. Paciﬁc J. Math. 30 (1969) 771775. 7. On mapping indecomposable continua onto certain chainable indecomposable continuum. Proc. Amer. Math. Soc. 25 No. 2 (1970) 449-456. 8. On compactiﬁcations with continua as remainders. Fund. Math. 9. Continuous mappings on continua. Proc. Auburn Topology Con- ference, 1969, Auburn University, Auburn, A abama, 94-97. 10. On mapping indecomposable continua onto certain chainable indecomposable continua. Proc. Amer. Math. Soc. 25 (1970) 449-456. 11. Decomposable inverse limits with a single bonding map on [0,1 below the identity. Fund. Math. 66 (1969/70) 177-183. 12. A space whose regions are the simple domains of another space. Comment. Math. Prace Math. 13 (1970) 141-159. 13. Continua that contains only degenerate continuous images of plane continua. Duke Math. J. 37 (1970) 479-483. 394 CHAPTER 9 Publications of Harvey L. Baker

Baker, Harvey L., Jr. 01 • 1. Complete amonotonic decompositions of compact continua. Proc. Amer. Math. Soc. 19 (1968) 847-853. t ACADEMIC DESCENDANTS AND PUBLICATIONS 395 Publications of J. W. Green

Green J. W. 01 • 1. Functions that are harmonic or zero. Amer. J. Math. 82 (1960) 867-872. 2. On the Arzela-Ascoli theorem. Math. Mag. 34 (1960/61) 199- 202. (with Valentine, F. A.) 3. Concerning the separation of certain plane-like spaces by compact dendrons. Duke Math. J. 37 (1970) 555-571. 396 CHAPTER 9 Publications of David E. Cook

Cook David E 01 • 1. A conditionally compact point set with noncompact closure. Pa- ciﬁc J. Math. 35 (1970) 313-319.

Publications of John W. Hinrichsen

Hinrichsen, John W. 01 • 1. Concerning upper semi-continuous decompositions of irreducible continua. Proc. Amer. Math. Soc. 30 (1971) 157-163. (with Transue, W. R.; Fitzpatrick, Ben)

Publications of Joel L. O’Connor

O’Connor Joel L. 01 • 1. Supercompactness of compact metric spaces. Nederl. Akad. Wetensch. Proc. Ser. A/3 = Indag. Math. 32 (1970) 30-34.

Publications of Robert E.Jackson

Jackson, Robert E. 01 • 1. Quasi-Jordanian continua. Proc. Amer. Math. Soc. 27 (1971) 387-390. Index

Aczel, J.* , 263 Bachelis, Gregory Frank , 295, 304, Addis, David , 311 308 Aitchison, Barbara , 223 Bacon, 11 Albeverio, S.* , 334 Bacon, Philip , 399–401 Alford, William Robert , 330, 332, Baildon, John D. , 311 379, 384 Bailey, J. R.* , 228, 337, 348, 361 Alzoobaee, Orabi H. , 395 Bailey, John Lay , 228, 337, 348, Anderson, Bruce A. , 392, 397 361 Anderson, David R. , 337, 346, Baker, Blanche , 180 347, 360 Baker, Harvey L. Jr. , 180, 411 Anderson, Lee W. , 224, 268, 271, Baker, John W.* , 277 276, 278–280, 284, 288 Ball, B. J., 112 Anderson, R. D., 112 Ball, Billy Jo , 180, 318, 333, 383, Anderson, Richard D. , 180, 287, 384 357, 367, 371, 373, 389, Bandy, Carroll L. , 404 395 Barlow, Richard E.* , 303 Andrews, James J. , 212, 225, 252, Barrett, Lida K. , 197 291 Barth, K. F.* , 334 Antonelli, Peter , 197, 198 Barton, M. V., 138 Archangelskii, A. V.* , 327 Bass, Charles D. , 338, 364 Arens, R. F.* , 239 Bass, Sam, 7 Armentrout, Steve , 180, 345, 356, Basye, R. E., 112 392, 393, 396 Basye, Robert Eugene , 180, 310 Arnquist, Cliﬀord Warren , 311 Batchelder, P., 87, 102, 138 Aronszajn, Nathan* , 195 Beal, 74, 84 Atkins, Anne, 11 Bean, Ralph , 337, 348, 360 Atticus, 19 Bear, Herbert Stanley Jr. , 225, Ayres, C. E., 157, 162 244, 290 Ayres, W. L., 66, 104 Beck, Oscar , 383 Ayres, William L. , 189, 228 Beckenbach, E. F.* , 345 Bednarek, A. R.* , 263 Babb, 74 Belﬁ, Victor , 338 Babcock, William Warren , 402 Bell, 74

397 398 CHAPTER 9

Bell, Curtis P. , 201 Boyd, William S. , 337 Bell, Harold , 402, 403 Boyer, Jean Marie , 223, 236, 237 Benedict, H. Y., 31, 33, 34, 101, Brahana, Thomas R. , 201, 250 102, 108, 124, 125, 136, BraSher, Rus , 223, 265 137, 172 Brauer, Alfred* , 305 Bennett, Albert , 337 Braun, J. , 386 Bennett, Donald Earl , 392 Brechner, Beverly L. , 367, 370, Bennett, Ralph B. , 384, 399, 400 371 Benton, Thomas C. , 192 Bredon, G. E.* , 257 Berg, Gordon Owen , 311 Brogan, 140, 148, 153 Bergman, John G. , 277 Broun, William LeRoy, 15, 23 Berkowitz, Harry W.* , 363 Browder, William* , 243 Bernard, Anthony Dwight Jr. , Brown, David L. , 224, 255, 281, 296 282, 285, 286 Berri, Manuel P. , 323–325, 328 Brown, Dennison R. , 224, 255, Beurling, Arne* , 302 281, 282, 285, 286 Bing, R. H. , 180, 212, 248, 337, Brown, E. W., 57, 89, 164, 165 340, 350, 359, 369, 393, Brown, Morton , 337, 349 396 Brown, T. A.* , 259 Blackwell, Paul , 198 Burdick, R. O.* , 247 Blakemore, Carol , 223 Burdine, A., 172 Blakley, G. R.* , 318 Burgess, C. E., 112 Blaschke, Wilhelm, 95 Burgess, Cecil Edmund , 180, 376, Boals, Alfred , 202, 221 378, 380, 381 Bohn, Elwood* , 325 Burke, 19 Boles, John Amos , 295 Burnam, 87 Bolyai, 25, 26, 33, 66, 72 Bushnell, Donald D. , 337 Bolza, Oscar, 52, 54 Butcher, George , 223, 264 Bompiani, Enrice, 95 Butler, Terence* , 299 Boner, C. P, 94, 125, 137, 138, 143, 148, 149, 156 Caesar, 19 Bookhout, Glen Allen , 296, 309 Cain, George L. , 225, 255, 292 Booth, Edwin, 8 Cajori, Florian, 33 Borel, 55, 63, 79 Calhoun, J. W., 8, 87, 101, 102, Boren, Nancy Susan , 402 108, 124 Borrego, Joseph T. Jr. , 224, 273, Cannon, James Weldon , 376, 377, 283, 289 380, 381 Borsuk, Karl* , 345 Cannon, Lawrence Orson , 376, Boswell, R. D. , 225 378 Bouricius, Willard G.* , 215 Cannon, Raymond Joseph Jr. , Boyce, William Martin , 330, 335, 330, 335 336 Cantrell, J. C.* , 363 ACADEMIC DESCENDANTS AND PUBLICATIONS 399

Capel, C. E.* , 265 Cohen, Leon W. , 207 Cargo, G. T.* , 335 Collins, H. S.* , 265, 276, 278 Carpenter, “Red”, 70 Connell, E. H.* , 369 Carrano, Frank M. Jr. , 295 Connelly, Robert , 337, 361 Carruth, J. H. , 224, 282, 285 Conner, Pierre E.* , 221, 247, 387 Cartier, Pierre* , 240 Conner, William , 223 Cash, Burt , 338 Connor, Andrew C. , 383 Casler, Burtis Griﬃn , 338, 354 Converse, George* , 243 Cato, 19 Cook, David Edwin , 180 Chadick, Stanley , 330 Cook, Howard , 180, 407, 408 Chae, Y. , 225, 275 Cooper, 19, 102, 138 Chambers, 74 Copeland, Arthur H. Jr.* , 221 Chapman, Thomas A. , 372, 373 Cornette, James L. , 180, 409 Charatonik, J. J.* , 286 Coven, Ethan M.* , 197 Chawla, L. M.* , 303 Cowan, Richard , 330 Cheeger, J.* , 361 Craggs, Robert F. , 338, 358, 363 Chew, Kim Peu , 323, 325, 328 Craig, H. V., 138, 163, 165 Chewning, William Carroll Jr., 409 Crawley, 73, 74, 85 Chicks, Charles H. , 295 Crittenden, Richard B.* , 269 Chittenden, E. W., 104, 105 Cross, Myrle V. Jr. , 202 Choe, Tae Ho , 224, 289 Crossley, S. Gene , 337 Choo, Eng Ung , 323, 325 Curtis, D. W. , 337, 360 Choquet, Gustave* , 369 Curtis, Morton L. , 207, 211, 219, Chrestenson, Hubert Edwin , 295, 220, 251, 252, 331, 345, 300, 302 352 Christoph, Francis T. Jr. , 311, Cutler, William H. , 338, 357 320 Cuttle, Percy Mortimer , 295, 303 Church, Philip T.* , 196 Cuttle, Yvonne G. , 295, 303 Cicero, 19 Civin, Paul , 295, 298, 302, 304 Dabney, 15, 25, 28 Clapp, Michael Howard , 225 Dancer, Wayne , 201 Clark, C. L. , 224, 225, 283, 286 Dancis, Jerome , 338 Cleveland, (President), 11 Darroch, J. N.* , 256 Cleveland, C. M, 93, 102, 108, 112, Daverman, Robert J. , 338, 364, 125 380 Cleveland, Clark Milton , 180, 309 Davis, Harvey S.* , 208, 209, 280 Cliett, Otis Jay , 402 Davis, Roy Dale Jr. , 180 Cobb, John I., III , 338, 357 Day, Jane Maxwell , 223, 263, 264, Cohen, Haskell , 223, 265, 276, 327 278, 337, 346 Decell, Henry P. Jr.* , 198 Cohen, Herman J. , 223, 265, 276, Decherd, George Michael, 44 278, 337, 346 Decherd, Henry Benjamin, 44 400 CHAPTER 9

Decherd, Kate Thompson, 44 Ettlinger, H. , 87, 91, 102, 125, Decherd, Mary E., 37, 38, 44, 87, 138, 139, 149, 160, 162, 102 163, 165, 166, 171, 172 Decherd, Mary E. , 38, 44, 87 Evans, Benny Dan , 338 Decherd, William Thompson, 44 Eves, Howard* , 246 deCoverly, Sir Roger , 19 DeGroot, J.* , 326, 328 Farley, Reuben , 224, 286 Dempsey, Jack, 168 Farmer, Frank Davis , 225 Denjoy, 97 Faucett, William M. , 224, 271 Detmer, Richard C.* , 322 Feichtinger, Oskar* , 358 DeVun, Esmond E. , 223, 283, 289 Feldman, Jacob* , 240 Dickman, R. F. Jr. , 209, 252, Fell, J. Michael Gardner , 225, 259, 293 239, 290 Dickson, Leonard Eugen, 26, 31– Ferguson, Edward N. , 224, 270 36, 42, 45, 49, 52, 54, Ferris, Ian , 338 56–58, 86 Fillmore, Jay P.* , 333 DiDonato, A. R.* , 305 Filmer, D. L.* , 335 Dieﬀenbach, Robert M. , 338 Finney, Ross Lee , 365, 366 Dillon, Richard Thomas , 295 Fischer, D. R. , 337 Dilworth, R. P.* , 268 Fisher, 32, 73, 74 Dimitroﬀ, G. E. , 224, 270 Fisher, Gordon McCrea , 367, 370 Dodd, E. L., 87, 89, 91, 102 Fiske, T. S., 57 Dolley, 158, 159 Fitzgerald, J. A., 125 Dorroh, J. L., 93, 112 Fitzpatrick, Ben T Jr.* , 385, 407, Dorroh, Joe Lee , 180, 297, 309 413 Dristy, Forrest* , 252 Fletcher, Peter* , 216 Dryden, 19 Flippen, W. H., 10 Duda, Edwin , 226, 257, 292, 293 Floyd, E. E. , 225, 246, 251, 291, Dugundji, James* , 212, 220 319, 340 Duhamel, 75 Ford, Jo W. , 318, 384 Dushnik, Ben* , 209 Forge, Auguste , 296, 306 Dyer, Eldon, 112 Franklin, Stanley P. , 263, 323, Dyer, Eldon , 180, 386, 388 327 Franklin, V.* , 239 Eaton, William Thomas , 358, 376, Fremon, Richard , 202 380 Friedberg, Michael , 224, 226, 281, Eberhart, Carl A. , 280, 282, 286 282, 285, 286, 400 Edmonds, 138 Fugate, J. Brauch , 392, 395 Edwards, Richard , 337, 361 Fuller, Richard V. , 255 Eilenberg, Samuel* , 210 Fulp, Ronald O.* , 285 Engelking, J. R.* , 317 Fulton, Lewis M. Jr. , 295, 305 ACADEMIC DESCENDANTS AND PUBLICATIONS 401

Gabai, Hyman* , 210 Hall, D. H* , 223, 236, 237, 239, Gallai, Tibor* , 195 260 Gary, John M. , 201, 217, 222 Hall, Dick Wick , 223, 236, 239, Gauld, David , 386, 388 260 Gehman, H. M., 104 Hallett, George H., 73, 74, 80, 179 Gehman, H. M. , 188 Hallett, George H. Jr., 80–83, 85, Gentry, Ivey Clinton , 295, 305 86 Gentry, Karl R. , 383, 384 Hallett, George H. Jr. , 200 Geoghagen, Ross , 338, 363 Halmos, Paul R.* , 210 Gerlach, Jacob , 338 Halsted, George Bruce, 23–27, 29– Gershenson, Hillel H. , 386, 387 35, 37–40, 42, 44, 45, 48, Gibbons, Joel , 225 49, 67, 81, 82, 85, 87, 98, Gilbert, Paul W. , 299 101, 110 Gillman, David , 338 Hamilcar, 19 Girolo, Jack E. , 409 Hamilton, O. H., 109 Glaser, Leslie C. , 338, 354, 356, Hamstrom, M. E., 112 362 Hamstrom, M. E. , 180, 369, 387, Gleason, A. M.* , 244 388 Glenn, 74 Hannibal, 19 Gluck, Herman* , 349 Haque, Mohammed Rashidul , 330, Goﬀman, Casper* , 207 333 Goldstein, Richard Z.* , 397 Harper, William Rainey, 49–51 Goodspeed, Thomas W., 49, 50 Harris, J. K. , 224 Gordh, George Rudolph Jr. *, Harry, C. H. , 223 311, 322 Haskew, 126, 127, 143, 144 Gordon, William L. , 223, 266 Haver, William E. , 311 Gottlieb, D. H.* , 277 Haynsworth, William Hugh , 226, Gould, Robert S., 15 292, 293 Grace, E. E. , 311, 316, 317 Heath, Robert Winship , 311, 316, Graham, 126, 127, 159 317, 320 Grant, U. S., 8 Greathouse, Charles A. , 201 Hedlund, G. A. , 251 Greef, Lynn George , 311 Helgason, Richard* , 222 Green, John William , 180 Hemmingsen, Erik , 196, 197 Greenwood, 138 Hempel, John P. , 338, 352, 354 Grilliot, Thomas J. , 201, 221 Henderson, David W. , 338, 356, Gropen, Arthur L. , 296 369, 372, 373, 396 Henderson, George W. , 180, 405 Hagopian, Charles L. , 311, 316, Henze, H. R., 138 320 Hermite, 32 Hahn, 99 Herrlich, H.* , 328 Hahn, Samuel Wilfred , 295, 304 Higliono, G.* , 326 402 CHAPTER 9

Hilber, 24, 38, 40–44, 48, 49, 55, Kelley, “Spider”, 70 62, 63, 71, 77 Kennedy, John F., 9 Hildebrand, S. K. , 337, 346, 347, Key, Margaret (Mrs. R. L. Moore), 360 73 Hill, C. Denson* , 335 Kirby, Helen M., 23 Hinman, Betty , 224 Klein, 51, 56 Hinrichsen, John W. , 180, 385, Kline, J. R, 79, 80, 85, 92, 94, 96, 413 104 Hobson, 79, 100 Klipple, E. C., 109, 112 Hocking, J. G.* , 219, 332 Kronech, 51 Hodel, Richard Earl , 296, 306, 308 Law, 14, 114, 127, 139 Hoggatt, Verner E. Jr. , 225, 245 Lee, Robert E., 15, 48, 68, 174 Holmes, 102 Lefschetz, S., 13, 72, 73 Honerlah, Raymond W. , 295 Lennes, N. J., 61–63 Hood, (General), 8 Lie, Sophus, 32–34, 36 Hoover, Herbert, 107 Lin, You-Feng , 224, 273, 289, 308 Horton, Goldie P., 87, 102 Lind, D. A.* , 246 Hosay, Norman , 338, 355 Lindahl, Robert J. , 295 Howell, L. B.* , 310 Lininger, Lloyd L. , 392, 394, 397 Hoyle, Hughes B., III* , 216, 385 Lister, Frederick M. , 379 Humphreys, Milton W., 15 Livesay, C. G.* , 357 Hunt, John H. V. , 293 Livingston, Arthur Eugene , 295 Hunter, Larry Clifton , 295, 303 Lobachevski, 25, 26, 33, 72 Hunter, Robert P. , 209, 272, 276– Lobb, Barry Lee , 296 278, 284, 288 Lock, 65 Hunziker, W.* , 334 Lomax, (Regent), 39 Husch, Lawrence S. , 225, 253, Lomonaco, S. J.* , 252 291, 363 Lorch, Lee* , 302 Hutchinson, Thomas C. , 392 Loveland, Lowell Duane , 327, 376, Hyman, Daniel M. , 337, 348 377, 381 Lowell, James Russell, 28 Iltis, Donald Richard , 295 Lubben, R. G, 90, 91, 93, 102, Jackson, Stonewall, 15, 25 108, 138, 139, 171, 172 Jaworowski, J. W.* , 264 Lubben, Renke G. , 180, 222 Jensen, Richrd A. , 326, 338, 364, Lusk, Ewing L. , 338 378 Lyon, Herbert C. , 338, 362 Johnston, (General), 8 Lyttle, R. A. , 225 Jones, F. Burton, 24, 41, 112 Jordan, 32, 64, 74, 76, 80 Macaulay, 19 MacDonald, H. M., 138 Kaplan, Samuel, 201, 210 Machigian, Jack* , 266 ACADEMIC DESCENDANTS AND PUBLICATIONS 403

MacKay, Roy , 201 McMillan, Evelyn R. , 226, 260, Madison, Bernard , 223, 280, 284– 294 286 McNamara, James , 386 Mahavier, William S. , 180, 282, McShane, E. J.* , 239 399, 400 Menger, Karl* , 195 Malbon, W. E. , 225 Merwin, R. E. , 201 Malcolmson, Waldemar, 10–13 Meyer, Donald V. , 392, 394, 395, Mann, L. N.* , 350, 388 397 Mann, W. R.* , 298 Meyer, Paul Andre* , 240 Manuel, 147 Michael, Ernest* , 317 Marciano, Rocky, 168 Michie, 102 Mardesic, Sibe* , 254 Middleton, D. P.* , 303 Martin, Abram Venable Jr. , 295, Miller, David L., 138–140, 152, 298, 299 153, 155–157 Martin, Joseph M. , 345, 359, 392, Miller, Don D. , 192, 198 394, 396, 402 Miller, Edwin W. , 201, 209 Marx, Morris Leon , 330, 333 Miller, Gary Glenn , 391 Maschke, Henrich, 52, 54, 58 Miller, H. C. , 112 Mason, William , 392 Miller, Harlan C. , 180, 329 Mathews, Harry Thomas , 330, Miller, John , 386, 388 333 Miller, Richard T. , 337 Mathews, J. C.* , 347 Milnes, Harold Willis* , 347 Maxﬁeld, John E.* , 303 Mishra, Arvand Mumar , 323 May, J. Gaylord , 383 Mislin, Guido* , 220 May, W. Graham , 383 Mislove, Michael W. , 224 Mayland, James , 392 Mitchell, 74, 81 McAllister, Byron L. , 311, 315, Mohat, John Theodore , 180, 390 338, 357 Mohler, Lee K. , 224, 271, 288 McArthur, C. W.* , 266 Moise, E. E., 112 McAuley, Louis F. , 276, 278, 311, Montgomery, Deane* , 194 319, 358 Moore, Charles, J., 8–10 McBay, Shirley M. , 201 Moore, Eliakim Hasting, 24, 26, McCharen, John D. , 224, 286, 36, 37, 39, 42–45, 48–51, 369, 371 53, 54, 56–63, 86, 99 McCord, M. C.* , 251 Moore, John T.* , 275 McCoy, R. A. , 337, 347, 360 Moore, Louisa Ann, 8, 9 McDougle, Paul E. , 257 Moore, R. L. , 188 McGuire, Carson, 138 Morava, Jack Johnson , 201, 220 McKinley, William S. , 337 Morgan, John W. , 201 McMillan, Daniel R. Jr. , 213, Morley, F., 26 338, 351, 353, 354, 361, Morris, Joseph Richard , 225, 256 362 Morse, Marston* , 384 404 CHAPTER 9

Moser, Louise , 392 Patterson, 96 Mozzochi, C. J.* , 270 Patty, C. Wayne , 201, 216, 221 Muenzenberger, T. R.* , 269 Payne, William W., 35 Muller, Herman Joseph, 96 Peano, 55, 62, 63 Mullikin, Anna, 83–85, 93 Pearson, Benny Jake , 180, 391 Mullikin, Anna M. , 179, 200 Penney, David Emory, III , 402, Mullin, J. B. , 330 403 Mullings, 102 Penniman, Josiah H., 79 Munkres, James R. , 365, 366 Peterson, Bruce B. , 196 Myer, 19 Peterson, Donald Palmer , 295, Myung, Myung Mi , 202 303 Pettey, Dix Hayes , 376, 382 Nadler, Sam B. Jr.* , 287 Phelps, R. R.* , 243, 290 Nagami, Keio* , 298, 299 Phillips, 32 Naimpally, S.* , 347 Picard, 32 Namioka, Isaac , 225, 239, 243, Pieri, 62, 63 290 Piranian, George* , 332 Nanzetta, Philip* , 293 Pitcher, Everett* , 239 Nashed, M. Z.* , 255 Pittman, Chatty R. , 383, 385 Nation, Carrie, 7 Plunkett, Robert L. , 255 Neuzil, John P. , 392, 398 Poe, R. L.* , 347 Newell, G. F.* , 256 Porter, Goldie P. Horton, 138 Newton, H. A., 26, 51 Porter, J. R.* , 328 Nichols, Joseph Caldwell , 296, 308 Porter, Milton Brocket, 30–32, 85, Nicholson, Victor A. , 338, 356 87, 101, 102, 138 Nolle, A. W., 138 Price, Thomas M. , 291, 338, 355, Nunnally, Ellard V. , 367, 370 362, 394 Nyikos, Peter , 323 Proﬃtt, Michael H. , 180 Propes, Ricahrd E. , 224, 270, 287 O’Brien, Thomas* , 199 Proschan, Frank* , 303 O’Connor, Joel Leslie , 180 Prosser, Reese Trego , 225, 290 O’Neill, James, 8 Prouse, 138 O’Steen, David , 367 Puckett, W. T. Jr.* , 237 Oden, Lynn* , 354 Purifoy, Jesse Allen , 180 Odle, John W. , 191 Putman, T. M., 32 Ollman, Loyal Frank , 191 Orlik, Peter* , 221 Quinn, Arthur Hobs, 79 Osgood, 75 Quinn, Joan , 202 Otto, H. H., 138 Ratti, J. S.* , 289 Pak, J.* , 216 Raymond, Frank A. , 201, 217, Pasch, 55, 62, 63 219, 257 ACADEMIC DESCENDANTS AND PUBLICATIONS 405

Reddy, William L. , 196–199 Roy, Prabir , 311, 317, 321 Reed, Bruce E. , 201, 216 Roy, S. M.* , 291 Reed, Dennis Keith , 180 Rozycki, Eugene P. , 360 Reed, James , 337 Rubin, Leonard R. , 209, 225, 252, Reed, Myra S. , 311 259, 291 Reinmuth, O. W, 138 Rudin, M. E., 112 Rhee, Choon Jai , 201, 216 Rudin, Mary E. Estill , 180, 374 Rice, 83, 95, 102 Rupp, 102 Rice, Peter Milton , 201 Rushing, T. B.* , 291, 363 Richardson, Joan , 392 Russell, Mary Jean , 399 Richardson, R. W.* , 248 Rutt, N. E., 104 Riecke, Carroll V. , 367 Rutt, Norman E. , 194 Riesz, F., 97 Riley, John Philip Jr. , 402, 403 Saadaldin, M. Jawad , 296, 306 Robbie, D. A. , 225 Sabello, Ralph , 323 Roberson, 87, 102 Saﬀord, 74 Roberts, J. H., 93, 108 Sanderson, Donald E. , 337, 346, Roberts, John Henderson , 180, 347, 360 295, 297, 300, 307–310 Scarborough, Charles T. , 323, 325, Roberts, Oran M., 15 326 Robertson, J. M.* , 378 Schlais, Harold Eugene , 311 Robertson, L. C.* , 328 Schmidt, D. L. , 337, 360 Robertson, Wendy* , 239, 243 Schmitt, 65, 66 Robinson, T. J. , 337 Schnare, Paul S. , 323, 327 Rockefeller, John D., 49, 50 Schnechenberger, Edith K. , 191 Rodriguez, Rene , 330 Schneider, Peter R.* , 215 Roeling, Lloyd G. , 338 Schneider, Walter Jan , 330, 336 Rogers, James Ted Jr. , 180, 222, Schoch, 156 311, 321, 410 Schoenﬂies, 61, 63, 99 Rogers, Leland Edward , 322 Schori, Richard M. , 357, 369, 392, Rolfsen, Dale P. , 392, 394, 397 395 Root, W. L.* , 243 Schrader, R.* , 334 Rose, David A. , 289 Schuh, Mark , 223 Rosen, Ronald H. , 338, 361 Schur, 43 Rosenbloom, P. C.* , 196 Schwartz, 51 Rosenstein, George Morris Jr. , Schwartz, A. J.* , 321 296, 298, 307 Schwatt, 32, 73, 74 Roth, John P. , 201, 207 Schweigert, G. E. , 235 Rothman, Neal J. , 223, 276, 277, Scott, D. S.* , 215 280, 284 Scott, W., 19 Rotman, Joseph* , 366 Secker, Martin D. , 180 Row, W. Harold , 338, 352 Segal, Jack , 225, 251, 253, 318 406 CHAPTER 9

Selden, John Jr. , 280, 283 Smith, Kenneth A. , 337 Selfridge, Ralph Gordon , 295, 302 Smith, Marion B. , 311 Shakespeare, 11 Smithson, Raymond E. , 224, 269, Shaler, Nathaniel Southgate, 9 287 Sharp, Henry Jr. , 295, 305, 308 Smythe, William R. , 295, 306 Shaw, Kay* , 305 Socrates, 170, 174 Sheldon, William L. , 224 Soniat, Leonard E. , 273, 296, 308 Sher, Richard Benjamin , 223, 265, Sorgenfrey, R. H., 112 333, 376, 379–381 Sorgenfrey, Robert H. , 180, 323– Sherman, (General), 8, 30, 50 325, 328 Shershin, Anthony Connors , 225, Spanier, E. H.* , 290 275, 289 Spence, Lawrence , 202 Shields, Allen* , 387 Spencer, Guilford L., II* , 237 Shirley, Edward D. , 289, 311, 322 Srinivasan, T. A.* , 290 Shoenfield, Joseph R. , 201, 220 Stadtlander, David P. , 209, 280, Shrader, Susan* , 198 285 Shub, Michael* , 257 Stafford, 102 Shuster, Seymour* , 251 Stafney, James D. , 295, 304 Siebenmann, L. C.* , 388 Starling, Greg , 223 Sigmon, Kermit N. , 224, 275, 289 Steenrod, N. E.* , 299 Silber, John R., 172, 173, 175 Steffenson, Arnold R. , 337 Simmons, 140 Stephenson, Robert M. Jr. , 323, Simms, D. J. , 388 325, 326, 328 Simon, Carl , 225 Stepp, James W. , 280, 285 Simon, Jonothan , 392 Stevenson, Nell Elizabeth , 180 Sims, B. T. , 337 Stewart, D. G.* , 396 Singh, Vashista N. , 225, 290 Stoddard, James H. , 202 Siry, Joseph W. , 223, 237 Stone, A. H.* , 314, 325 Slack, Stephen P. , 338 Stone, E. A. , 226 Slaught, H. E., 57 Stone, Ormond, 50, 51 Slaughter, Frank Gill Jr. , 296, Strebe, David D. , 189 299, 307–309 Strecker, George E. , 323, 325, Slover, R.* , 360 328 Slye, J. M., 112 Strocchi, Franco* , 251 Slye, John Marshall , 180, 390 Strohl, George Ralph , 237 Smith, Edgar F., 74, 79 Strong, 32, 53 Smith, F. H., 35 Strother, Waymon L. , 223, 265, Smith, Jack Warren , 226, 296, 269 308 Strube, Richard , 337 Smith, James Clarence Jr. , 226, Su, Li Pi , 323, 324, 328 296, 308 Summerhill, R. Richard , 392, 398 Smith, Keenan T.* , 337 Supnick, Fred* , 346 ACADEMIC DESCENDANTS AND PUBLICATIONS 407

Swain, R. L., 112 Vickery, C. W., 109, 112 Swain, Robert L. , 180 Vickery, Charles Watson , 180, 310 Sylvester, J. J., 27–29 Virgil, 19 Vobach, Arnold R. , 367, 371 Taft, Horace, 50 Vought, Eldon Jon , 314, 318, 321 Taft, William Howard, 50 Voxman, William L. , 392, 397 Tallichet, H., 15 Tarski, Alfred* , 240 Wade, L. I.* , 266 Taylor, 66 Waggener, Leslie, 15 Taylor, T. U., 25, 29 Wagner, E. G.* , 215 Teh, H. H.* , 325 Wagner, Robert P., 138 Tennyson, A., 19 Wall, H. S, 47, 68, 138, 139, 162, Thatcher, Sherman, 50 171, 172 Thoma, Elmar* , 241 Wallace, Alexander Doniphan , 223, Thomas, Edward Sandusky Jr. , 260, 276, 278, 283, 328 254, 311, 314, 318, 321, Walsh, Bertram* , 244 384 Walsh, J. L.* , 336 Thomas, Garth , 337, 346 Ward, Lee E. Jr. , 224, 265, 266, Thomas, Joseph, 11, 17, 49, 50, 271, 272, 287, 367 74, 83–85 Wardwell, James F. , 236 Tindell, Ralph S. , 225, 253 Warren, Nancy , 374 Titus, C. J.* , 332 Wattel, E.* , 326, 328 Tomlinson, Michael B. , 295 Wayment, Stanley G.* , 380 Traylor, D. Reginald*, 404 Webster, Dallas E. , 338 Treybig, L. Bruce , 180, 402, 403 Weierstrass, 51 Tucker, L. D. , 224 Weiner, 13 Tulley, Patricia A.* , 315 Weiss, Max* , 244 Turner, Edward , 386, 388 Wells, Carroll Glenn , 330 Tymchatyn, E. D. , 269, 271, 288 Wenner, Bruce Richard , 296, 307, 309 Ungar, Gerald S. , 311, 315, 319 Werremeyer, Frederic N. , 337 Wertheimer, Stanley J. , 225 Valentine, F. A.* , 412 West, James Edward , 357, 367, Van Cleave, John T. , 399 369, 372 Vance, Elbridge P. , 191 Weyl, Herman, 95 Vandiver, H. S., 91, 102, 137, 138 Wheeler, Charles, III , 223 Vaughan, Herbert E. , 210 White, Christopher Clark , 295, Vaughan, Jerry E. , 296, 307, 309 300, 304 Vaught, R. L.* , 239 White, J. A., 107, 138, 150 Veblen, Oswa, 54, 55, 60–64, 67, White, Paul A. , 225 68, 75–77, 97 White, Warrn , 392, 394, 397 Verhey, Roger F.* , 333 Whittaker, James V. , 323, 324 408 CHAPTER 9

Whyburn, Gordon T., 90–94, 104, Zeeman, E. C.* , 213, 353 134 Zippin, Leo, 104 Whyburn, Gordon T. , 180, 190, Zippin, Leo , 192 223, 227, 237, 283 Zoretti, L., 96, 97 Wicke, H. H.* , 406 Widder, 51 Wiginton, C. Lamar , 198 Wilder, R. L., 89, 92, 93, 104, 167 Wilder, Raymond L. , 180, 201, 203, 213, 215 Wilkinson, James B. , 296 Williams, 84, 160 Williams, Jerry F. , 399 Williams, Louis , 295 Williams, Robert F. , 256, 292 Williams, W. W. , 224, 285, 286 Wilson, David C. , 311, 319 Wilson, Logan, 68 Wilson, Woodrow, 67, 139 Wiser, Horace Clark , 376, 377 Wong, Raymond Yen-Tin , 367, 371 Woodruﬀ, Edythe P. , 311, 320 Worrell, John M. Jr. , 180, 406 Wright, Alden H. , 338, 362 Wright, Perrin T. , 338, 358, 363 Wright, Reverdy , 295 Wu, Chien-Heng* , 256 Wyss, Orville, 138 Xenophon, 19 Yeung, Henderson C. H. , 224 Yood, Bertram* , 301 Young, 31, 33, 97 Young, Frederick Harris, 295, 301 Young, Gail S., 112 Young, Gail S. , 180, 213, 330, 331, 335 Younglove, James N. , 180, 404 Youngs, J. W. T.* , 237 Zaccaro, Luke N. , 295