THE FIXED-POINT QUESTION FOR BOUNDED NON-SEPARATING PLANE CONTINUA By SISTER JOANN LOUISE MARK I\ Bachelor of Arts Kansas Newman College Wichita, Kansas 1962 Master of Science Oklahoma State University Stillwater, Oklahoma. 1968 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF EDUCATION May, 1975 OKLAHOMA STATE UNIVERSITY UBRARY MAY .12 1976 THE FIXED-POINT QUESTION FOR BOUNDED NON-SEPARATING PLANE CONTINUA Thesis Approved: D~of the Graduate College 9389G3 ii PREFACE Mathematics is more than a discipline. It is a living and de­ veloping art intertwined with the lives of people. It is my hope that Chapter II of this thesis preserves and exposes this exciting aspect of the fixed-point question in such a way that it is accessible to both graduate students in mathematics and to teachers of undergraduate mathematics. As an art, mathematics has certain fundamental techniques. In particular, the area of fixed-point theory has its specific artistry. Chapter III is my effort to expose the basic techniques in fixed-point theorems. While writing this thesis, it has been my pleasure to meet some of the great artists in fixed-point theory. I want to express to them my appreciation for any time or interest given to this thesis. I wish to express my particular appreciation to my major adviser, Dr. John Jobe, who shared with me his view of and his artistry in mathematics, enabling me to successfully complete this thesis. Es­ pecially I thank him for his friendship which has encouraged and supported my personal and mathematical development. I also owe thanks to Dr. James Choike, Dr. Douglas Aichele, and Dr. Ronald McNew. As members of my committee, they provided me with just the right mixture of challenge and encouragement. iii To the members of my family, to my friends, and to my religious community, The Adorers of the Blood of Christ, I also express my thanks. In particular I thank Sister Remigia Kerschen for her oontinuing support and Sister Dolores Strunk for her typing of the final copy of the thesis. My graduate study was supported financially through a Title III Grant awarded to Kansas Newman College, Wichita, Kansas. Consequently, I wish to thank the administration and faculty at Kansas Newman College who provided this support, and who encouraged me in my endeavor. iv TABLE OF CONTENTS Chapter Page I. INTRODUCTION • • • • • • • • • • • • • • • • • • • • • • l Topic of the Study • • • • • • • • • . l Procedure • • • • • • • • • • • • • • • • • • • • • 4 Preliminary Definitions ••••••• • • • • • • • 5 II. HISTORY OF THE FIXED-POINT Q'tJESTION • • • • • • • • • • 7 Barly Years: 1925-1937 • • • • • • • • • • • • • • · 7 The Years 1938-1950 • • • • • • • • • • • • • • • • 13 The Years 1951-1964 • • • • • • • • • • • • • • • • 16 The Years 1965-1973 • • • • • • • • • • • • • • • • 25 Present Efforts • • • • • • • • • • • • • • • • • • 35 III. FUNDAMENTAL TYPES OF PROOFS • • • • • • • • • • • • • • 38 The Dog-Chases-Rabbit Technique • • • • • • • • • • 38 The Immediate Technique • • • • • • • • • • • • • • 47 The Cyclic Element Technique • • • • • • • • • • • 57 The Change of Topology Technique • • • • • • • • • 61 The Sequence of Arcs Technique • • • • • • • • • • 64 Swnmary and Conclusions • • • • • • • • • • • • • • 66 IV. AN EXPOSITION OF BELL'S PAPER AND DEPENDENT RESULTS • • 68 Introduction • • • • • • • • • • • • • • • • • • • 68 Basic Understandings • • • • • • • • • • • • • • • 70 Preliminary Theorems • • • • • • • • • • • • • • • 80 Implemeni;ation • • • • • • • • • • • • • • • • 85 Conclusion of the Proof of Theorem 2.14 • • • • • • 88 Dependent Results • • • • • • • • • • • • • • • • • 96 v. SUMMARY • • • • • • • • • • • • • • • • • • • • • • • • 98 A SELECTED BIBLIOGRAPHY • • • • • • • • • • • • • • • • • • • • 101 APPENDIX A - PROOFS • • • • • • • • • • • • • • • • • • • • • • 105 APPENDIX B - INDEX OF TERMS • • • • • • • • • • • • • • • • • • 127 v LIST OF FIGURES Figure Page 1. Dendrite • • • • • • • • • • • • • • • • • • • • • • • • • 9 2. Unicoherent Continuum not Locally Connected • • • • • • • • 10 3.. Decomposable Boundaries • • • • • • • • • • • • • • • • • • 14 4. Indecomposable Continuum • • • • • • • • • • • • • • • • • 14 5. Pseudo-Arc • • • • • • • • • • • • • • • • • • • • • 19 6. Arcwise Connected Continuum. Without Fixed-Point . 24 Case (iii) of Theorem 2.1 . 39 8. The Set nd n . 49 Example 11 . 59 10. Example 12 • • • • • • • • • • • • • • • • • • • • • • • • 59 11. Illustration of Definitions 4.1-4.4 . 72 12. The set f (x) • . • • . 77 0 13. Illustration of Theorem 4.7 . 81 14. The Set H ,. J n Y 88 n n . vi CHAPTER I INTRODUCTION Topic of the Study The topic of this thesis is a question which has intrigued and baffled mathematicians for decades: "Does an arbitrary bounded plane continuUA which does not separate the plane have the fixed-point property?" Outstanding mathematicians have spent years trying to de­ finitively answer this question. Through their efforts they have proved many directly related theorems, as well as partial solutions to the original problem. It is the purpose of this paper to gather to­ gether these results, to present them along with interesting comments in chronological order, and to analyse the techniques used in their proofs. As a basis for understanding the topic under discussion the terms bounded, continuum, separate the plane, and fixed-point property will be defined. Definition 1.1. A subset M of the plane is said to be bounded with respect to the metric p if and only if M • ¢ or there exists a real number r such that p (x,y) ~ r for all x,y in M. In this thesis all sets will be situated in the plane unless otherwise specified. The p of Definition 1.1 will be the usual metric in the plane. 1 2 Before defining continuum, it is necessary to present several preliminary definitions. Definition 1.2. A point p is a ,!!!!!! point of a met M if and only if every open set containing p contains a point q & M, q ~ p. In the plane countable compactness and compactness are equiva­ lent. Consequently, the following definition will be used for the definition of compactness. Definition 1.3. A point set M is said to be compact if and only if every infinite subset of M has at least one limit point in M. Definition 1.4. Two subsets A and B of a space S are said to be sepa­ rated, denoted sep., if and only if A I-¢, BI-¢, An B. ¢, An B. f6. Definition 1.5. A subset M of a space S is connected if and only if M is not the union of two separated sets. Using the concepts of Definitions 1.2-1.5, a continuum can be de­ fined in the following concise terms. Definition 1.6. A continuum is a compact connected space. Let the continuum of the question "Does an arbitrary bounded plane continuum which does not separate the plane have the fixed-point property?" be denoted by M. The set M is given to be bounded. However, by the Heine-Borel Theorem in 2-space, it is also closed. Other theo­ rems from Hall and Spencer (18) and properties of the plane, imply M is also 1) T1, 2) T2, or Hausdorff, 3) let countable, 4) 2nd countable, 5) regular, 6) normal, 7) metric, and 8) separable. The definitions of these properties can be found in Hall and Spencer (18) or any elementary topology book. A special property of M is that it does not separate the plane. An abundant supply of examples of such continua will be given in Chapter II. It is necessary at this time to define this characteristic. Definition 1.7. The point set M is said to separate]!!! plane P if and only if MC P and M - P • A U B sep. From Bing (5) any plane continuWD which does riot separate the plane is the intersection of a decreasing sequence of topological disks. That is, M is the intersection of a decreasing sequence of sets homeo­ morphic to the set { (x,y)/ x2 + y2 s 1 } • Definition 1.8 will clari­ fy the meaning of homeomorphic. Definition 1.8. Let S and T be spaces. Then S is said to be homeo- morphic to T if and only if there exists a one-to-one open continuous mapping of S onto T. The mapping is called a homeomorphism. The question whether or not M has the fixed-point property is a question regarding the behavior of continuous functions f : M ~ M. Be- fore stating the fixed-point property, continuous will be defined. Definition 1.9. Let S and T be spaces and fa S -> T a function. Then tis said to be continuous at the point s CS if and only if, given &IJ1' open subset G of. T such that s c f-1(G), there exists an open set V of S such thats c VC f-1(G). It is now possible to define the pivotal concept of this thesis, the fixed-point property. Definition 1.10. A topological space S is said to have the fixed- point property (fpp) if, given any continuous function f from S into S, there exists a point p such that f(p) • p. During the remainder of this thesis, the fixed-point question or the fixed-point problem will refer to the question asked at the be- ginning of the chapters "Does an arbitrary bounded plane continuum which does not separate the plane have the fixed-point property?11 • As was mentioned earlier, the setting will be the plane unless otherwise stated. One basic and famous fixed-point result will be assumed, name- ly, the Brouwer Fixed-Point Theorem. Theorem 1.1. 0 s_ x1 $.1, for i • 1,2, ••• ,n } • Let f: In ->In be continuous. There is a point z in In such that f(z) • z. Procedure Chapter II contains the important dates and theorems in the study of the fixed-point question. These are arranged in chronological order and interspersed with both examples and related comments. The purpose of this chapter is to identify and correlate important fixed-point results. All major theorems stated in Chapter II except those of Harold Bell and Charles Hagopian have their proofs analyzed in Chapter III. Different titles are attached to different techniques of proof. Common and distinguishing elements are discussed and illustrated.