http://dx.doi.org/10.1090/coll/032

AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME 32

Topology of Manifolds

Raymond Louis Wilder

American Mathematical Society Providence, Rhode Island International Standard Seria l Numbe r 0065-925 8 International Standar d Boo k Number 0-8218-1032- 4 Library o f Congress Catalog Card Numbe r 49-672 2

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given. Republication, systematic copying, or multiple reproduction o f any material i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mail t o reprint-permissionQmath.ams.org .

Copyright © 1949 , 196 3 by the American Mathematica l Societ y Revised edition, 196 3 Revised edition , fourt h printing , with corrections, 197 9

The America n Mathematica l Societ y retain s al l right s except thos e grante d to the United States Government . Printed i n the Unite d State s of America . @ Th e pape r use d i n this book i s acid-free an d fall s withi n th e guideline s established to ensure permanence and durability .

10 9 8 7 6 0 1 0 0 9 9 9 8 TABLE O F CONTENT S

PREFACE . vi i

INTRODUCTION T O THE 196 3 EDITIO N x i

NOTES T O THE 196 3 EDITIO N x i

NOTES T O TH E 197 9 PRINTIN G xii i

I. ELEMENTAR Y CONCEPTS ; CHARACTERIZATION S O F E l AN D S l 1 1. Set s 1 2. Space s 2 3. Metri c space s 4 4. Close d and open subsets o f a space 5 5. Mappings ; homeomorphisms 6 6. Historica l remark s 1 0 7. Connecte d space s 1 6 8. Components ; quasi-components 1 8 9. Connecte d spaces satisfying the 2nd Hausdorff axio m and the weak separation axiom 1 9 10. Space s irreducibly connected abou t a subset 2 1 11. Th e simple ar c and the 1-spher e 2 7 12. Som e fundamental lemma s 3 3

IL LOCALL Y CONNECTED SPACES ; FUNDAMENTAL PROPERTIES OF THE EUCUDEAN TI-SPHER E 4 0 1. Loca l connectedness 4 0 2. Irreducibl e lc-connexes; recognition o f E l an d S 1 amon g lc spaces ... . 4 2 3. Som e general properties o f lc spaces . 4 5 4. "Phragmen-Brouwe r properties " and their equivalence in lc spaces .... 4 7 5. Som e topology o f th e n-spher e 5 1

III. PEAN O SPACES ; CHARACTERIZATION S O F S* AN D TH E 2-MANIFOLD S 6 9 1. Pean o continua ; mapping theorems 6 9 2. Topologica l characterizatio n o f Peano continu a 7 4 3. Pean o spaces 7 6 4. Recognitio n o f the 2-spher e 8 7 5. Recognitio n o f th e close d 2-cell 9 2 6. Recognitio n o f the 2-manifolds . . . • 9 4

IV. NON-METRI C L C SPACES, WIT H APPLICATION S T O SUBSET S O F TH E 2-SPHER E .. . 9 9 1. Component s o f locall y compac t Hausdorf f space s 9 9 2. A characterization o f locally compact, connecte d space s that fai l to be l c 10 2 3. Som e characterizations o f locally compac t l c spaces 10 4 4. Relation s between lc , S and ulc properties 10 7 5. Accessibilit y 11 0 6. Mor e properties o f the 2-sphere 11 2 7. Recognitio n o f Pean o continua i n S* by accessibilit y propertie s 11 6 8. Remark s 11 8

iii iv TABL E O F CONTENT S

V. BASI C ALGEBRAI C TOPOLOG Y 12 0 1. Complexe s 12 0 2. Algebrai c apparatu s 12 0 3. Chai n groups 12 2 4. Homolog y group s 12 3 5. Importan t specia l case s and geometric interpretation s 12 4 6. Som e fundamental lemma s 12 6 7. Cec h cycles and homolog y group s 12 9 8. Coverin g lemma s 13 3 9. Vecto r spaces 13 5 10. Existenc e theorem s 13 8 11. Som e applications t o connectedness an d local connectednes s 14 1 12. Fundamenta l system s o f cycle s fo r a compact metri c spac e 14 5 13. Alternativ e definition s 14 6 14. Dua l homomorphism s 14 8 15. Cocycles ; cohomology group s 14 9 16. Chai n product s fo r a complex 15 3 17. Extensio n to topological space s 15 6 18. Scala r products and dua l pairing s 15 9 19. Application s to homology propertie s o f spaces 16 7 20. Homologie s i n non-compact space s 16 8 21. Approximat e homologie s . 17 2

VI. LOCA L CONNECTEDNES S AN D LOCA L CO-CONNECTEDNES S 17 6 1. Loca l connectedness in dimensio n n 17 6 2. Chain-realization s 17 7 3. Complex-lik e characte r o f compac t lc n spaces 18 0 4. Non-compac t case s 18 3 5. Fundamenta l system s o f cycle s • 18 5 6. Loca l co-connectedness; local connectivity number s and local dualities . . 18 9

7. Propert y (P , Q) n , . 19 3 8. Othe r types o f highe r dimensional loca l connectednes s 19 8

VII. APPLICATIO N OF HOMOLOGY AND COHOMOLOG Y THEORY T O THE THEORY OF CONTINUA 20 0 1. Fundamenta l lemma s 20 0 2. Existenc e lemma s regardin g carrier s of cycle s and homologies 20 4 3. Separation s o f continu a by close d subsets 21 1 4. Non-r-cu t an d r-avoidable point s 21 8 5. r-extendabilit y 22 4 6. Non-cut-point s an d avoidable point s 22 7 7. Propert y £„ 23 4 8. Set-avoidabilit y 23 8 9. A n additio n theore m 24 1

VIII. GENERALIZE D MANIFOLDS ; DUALITIE S O F THE POINCAR E AN D ALEXANDE R TYP E . . 24 4 1. Genera l propertie s 24 4 2. Orientabilit y 24 6 3. Th e orientabl e n-gc m 25 0 4. Th e Poincar S dualit y fo r an orientabl e n-gc m 25 2 5. Th e ope n n-gm 25 4 6. Th e Alexande r type o f dualit y fo r a closed subset o f a n n-gcm. Firs t proo f 26 1 7. Th e Alexander type of duality for a closed subset of an n-gcm. Secon d proof 26 3 8. Linkin g theorem s 26 6 9. A dualit y fo r non-closed set s 26 9 TABLE O F CONTENT S v

IX. FURTHE R PROPERTIE S O F n-GMs ; REGULA R MANIFOLD S AN D GENERALIZE D TI-CELL S 27 1 1. Cas e n = 1 27 1 2. Cas e n = 2 27 2 3. Avoidabilit y propertie s 27 2 4. Characterizatio n b y mean s o f loca l linking 27 4 5. Th e cas e n = 2 without th e orientability conditio n 27 5 6. Th e general non-orientable cas e 28 0 7. Compariso n o f the case n > 2 with the classical case; regular manifold s an d generalized n-cell s 28 2

X. SUBMANIFOLD S OF A MANIFOLD; DECOMPOSITIO N INT O CELL S 29 0 1. Positiona l invariants 29 1 2. Unifor m loca l co-connectedness; duality o f r-ul c and ( n — r)-coulc .... 29 2 3. Th e Jordan-Brouwer separatio n theore m i n an n-gcm , an d it s converse . . 29 4 4. Generalizatio n 29 6 5. Additiona l positiona l propertie s 29 8 6. Th e boundary o f a ulc n~2 domain i n a manifol d 30 4 7. Additiona l converse s o f the Jordan-Brouwer separatio n theore m 30 7 8. Th e genera l ulc n~2 open subse t o f an n-gc m 30 9 9. Decompositio n o f the spherelik e n-gc m int o two generalized close d n-cells . 31 1

XI. Lc * SUBSET S OF A N n-G M 31 6 1. Dualit y o f th e S properties 31 6 2. Dualit y betwee n l c and S properties 32 0 3. Dualit y wit h S properties in terms o f cohomolog y 32 7 4. Relatio n o f avoidability propertie s at a point to S properties o f the comple - ment o f a close d set 33 1 5. Wea k S properties; recognition o f lc* boundaries from properties o f the domain 33 6 6. Wea k unifor m loca l connectedness 34 4 7. L c sets whose complementary domain s ar e bounded b y manifold s ... . 34 6

XII. ACCESSIBILIT Y AN D IT S APPLICATIONS 35 3 1. Regula r r-accessibilit y 35 3 2. Stronge r types o f accessibility; their interrelations and topologica l in variance 36 1 3. Application s to recognitio n o f submanifolds o f a manifol d 37 2

APPENDIX. SOM E UNSOLVE D PROBLEM S 38 1 1. Poin t set problem s 38 1 2. Problem s concernin g homolog y 38 1 3. Dimensio n theor y problem s 38 2 4. Problem s concernin g generalize d manifold s 38 2

BIBLIOGRAPHY 38 5

INDEX O F SYMBOL S • 39 3

INDEX 39 6

AUTHORS CITE D 40 2

F.RRATA 40 3 This page intentionally left blank PREFACE

The historica l backgroun d o f thi s wor k i s sketche d i n Chapte r I , sectio n 6 , and need not be repeated here. I t should, however, be complemented by certai n remarks o f a more personal nature, particularly as regards the author's indebted - ness to his mathematical colleagues . It ha s become more or less apparent t o students o f cultural evolution that th e genesis o f a lin e o f though t canno t b e fixe d eithe r i n chronologica l fashio n o r bibliographically. I f prope r evidenc e wer e o n record , a n ide a whic h seem s t o emanate a t a fixed dat e o r i n a particula r wor k woul d b e foun d upo n analysi s to be only the end product o f a collectio n o f prio r ideas; the "originator " o f th e idea bein g onl y th e mediu m throug h whic h thes e latte r idea s achiev e thei r synthesis. Eve n th e particula r individualit y o f th e "originator " i s probabl y not o f paramoun t importance ; o f importanc e i s th e perennia l presenc e o f th e "creative" mind , read y t o receiv e th e stimuli . Ca n anyon e doub t tha t th e calculus woul d hav e evolve d eve n thoug h Leibnit z an d Newto n ha d take n u p farming instea d o f science ? Simultaneou s announcemen t o f "discoveries " b y contemporaries, often widel y separated, is not a rare occurrence. It i s fitting , then , fo r a n autho r t o attemp t t o plac e hi s wor k i n it s prope r setting amongs t pas t an d contemporar y influences . Thi s i s th e objec t o f th e historical remarks i n Chapter I . Bu t thes e forma l remark s onl y partially fil l i n the picture . O n th e mor e persona l side , I wis h t o expres s m y indebtednes s t o Professor R . L . Moore , unde r whos e tutelag e I receive d a thoroug h groundin g in point se t theory . I t wa s during my earl y contact s wit h hi m tha t I cam e t o realize the vacuu m i n our knowledg e o f the set-theoreti c structur e o f the n-cell , particularly th e lac k o f a topologica l characterization . Later , throug h persona l contacts wit h Professo r Pau l Alexandrof f i n 1928 , I becam e convince d ( a con - viction which he obviously shared) that th e problem o f the n-cel l demanded ne w tools, especiall y th e extensio n t o genera l space s o f th e theor y o f connectivit y (homology). Acknowledgement s ar e also due to Professor Eduar d Cec h (whos e theory o f general homology is used herein), who visited the United States in 1934 - 35 and fro m who m I gaine d muc h stimulatio n an d persona l encouragement . I am also grateful t o the Institute fo r Advanced Study for making possible a year' s uninterrupted researc h i n 1933-34 , durin g whic h th e presen t investigation s o n manifolds wer e initiated; and to the John Simon Guggenheim Memoria l Founda - tion fo r th e gran t o f a fellowshi p i n 1940-41 . I t wa s durin g th e latte r perio d that th e euclidea n for m o f man y o f th e result s give n i n Chapter s X-XII wer e found. As regard s th e en d result—th e boo k itself—i t canno t b e emphasize d to o strongly tha t wha t i s presented herewit h i s only a beginning . I t i s onl y thos e properties o f manifold s tha t ca n b e handle d b y set-theoreti c an d homologi c vii Vlll PREFACE tools that ar e developed , an d eve n thes e ar e not completel y treated . Problem s concerning homotopy, mappings o f manifolds, applications to the study o f grou p manifolds, etc. , ar e al l awaitin g attention . Bu t I hop e tha t wha t i s don e her e will serve as a useful basi s for a n attack o n such problems. The delay i n publishing has been due to several factors. Sinc e my deliver y of the Colloquiu m Lecture s o n "Topolog y o f Manifolds " a t Vassa r i n September , 1942, in whic h the genera l outline s o f thi s wor k wer e presented, th e majo r par t of a war has been fought, an d a teacher in American universities need not be told what th e attendin g demands , an d th e heav y post-wa r universit y enrollmen t o f veterans, have don e to th e time that ca n be devoted to research. Also , most o f the result s i n th e late r chapters , publishe d her e fo r th e first time , wer e worke d out wit h th e euclidea n n-spac e a s locale . Resettin g thes e i n th e generalize d manifolds require d no t onl y revampin g o f proof s bu t takin g advantag e o f th e parallel advance s i n algebrai c topology . Ne w an d mor e powerfu l tool s wer e developing, suc h a s the theor y o f cohomolog y an d chai n products, whos e incor - poration necessitated much revision but which justified themselve s by the greater simplicity mad e possibl e i n proofs . I n man y cases , proofs involvin g homolog y which wer e lon g an d difficul t becam e muc h simplifie d throug h th e devic e o f reverting to cohomology . It als o becam e apparen t tha t th e wor k woul d hav e t o b e topologicall y self - containing; th e reade r coul d not b e expected to hav e previousl y rea d work s o n point se t theory , topolog y o f polyhedral s (combinatoria l topology ) an d th e newer algebraic topology. O n the other hand, it was not possible to write a com- plete expositio n o f al l these aspect s o f topology . Th e pla n finall y adopte d wa s to develop the program fro m its simplest elements to its more complicated stage s while simultaneously introducing the tools needed. Startin g at first with general spaces, sufficien t topologica l propertie s ar e introduce d t o characteriz e th e basi c 1-dimensional configuration s (arc , 1-sphere) . A s a consequence , Chapte r I i s quite elementary . Som e o f th e Schoenflie s result s i n tw o dimension s ar e the n given a s wel l a s som e o f th e mor e moder n plan e poin t se t theory—partl y t o furnish a natura l basi s an d motiv e fo r th e n-dimensiona l cas e an d partl y t o present a unified treatmen t whic h takes advantage o f the newer methods. Algebraic topolog y i s not introduce d unti l needed—som e topolog y o f polyhe - drals enters incidenta l t o th e materia l o n the euclidea n n-spher e i n Chapte r II , the mor e recen t algebrai c topolog y not bein g introduced unti l Chapte r V . Al - though the treatment o f these topics obviously could not be made in such general and complet e fashion a s in the companion volume by Lefschetz [L ] in this series, enough i s given to carr y through the later chapters . Th e discernin g reader wil l see many algebrai c problems to be solved. Throughou t th e late r chapter s onl y an algebrai c field i s used a s coefficien t group , since , fo r example , th e geometri c form o f the Alexander-Pontrjagi n dualit y form s a n important too l (thre e coeffi - cient groups are usually involved—one to defin e the manifold, an d on e eac h for * the homolog y theor y o f a subset M an d fo r th e complemen t o f M). However , it i s impossible t o d o more i n a wor k o f thi s siz e than t o sketc h i n the genera l PREFACE IX picture; the author hopes that other writers will fill in some o f the gaps and brin g the picture int o sharper focus . In a n Appendix , I have pointe d ou t som e unsolve d problems . Som e o f thes e may hav e ver y simpl e solutions ; others (a s fo r instanc e 1.1 ) ar e probabl y quit e difficult. Suc h well-know n (an d difficult ) classica l problem s as the classificatio n of manifolds , condition s unde r whic h th e S 2 i n S 3 bound s a 3-cell , etc. , ar e omitted. References t o th e bibliograph y ar e enclose d i n brackets , thos e involvin g capital letter s suc h a s [V ] o r [Mo ] referrin g t o book s o n topology , an d thos e involving onl y lower case letters suc h a s [a] , [c] referring t o miscellanea, mainl y journal articles . Pag e numbers, etc., may b e included, a s in [a ; 20] referring t o page 2 0 o f the articl e cited . Cross-reference s t o item s i n the tex t ar e generall y made b y citin g chapte r an d section ; thus " V 12.2 " refers t o Chapte r V , sectio n 12.2. Whe n a sectio n numbe r alon e occurs , suc h a s "12.2" , the referenc e i s t o the chapter in which the citation occurs . Reference s to formulae ar e enclosed in parentheses. Along with the index o f terms, there i s included fo r easy reference a n index of symbols. Certai n symbol s whic h refe r t o analogou s concept s migh t easil y b e confused. Th e latte r remar k applie s particularl y t o th e symbol s fo r homolog y and cohomolog y groups . Th e proble m o f symbolizin g th e variou s type s o f these group s whic h ar e encountered i n the presen t work , an d th e correspondin g Betti numbers , prove d a seriou s one , an d i t i s questionable i f i t ha s bee n satis - factorily solved ! I a m grateful t o those who have lent their advice, read some o f the chapters or assisted i n readin g proofs ; particularl y t o Professor s Miria m C . Ayer , E . G . Begle, S . Kaplan , P . A . Whit e an d Gai l S . Young; als o to Dr . K . E . Butcher , Dr. E. H. Larguier and Messrs. M. L. Curtis and L. F. Hsieh. Ai d in preparation of th e manuscrip t wa s receive d fro m th e Alexande r Ziwe t Fund , administere d by th e Executiv e Boar d o f th e Rackha m Schoo l o f Graduat e Studie s o f th e University o f Michigan . I wis h to thank the American Mathematical Society for the honor and privileg e of publishing this volume i n its Colloquiu m series . Ann Arbor, Michiga n December, 194 8 This page intentionally left blank INTRODUCTION T O TH E 196 3 EDITIO N

This editio n represent s primaril y a reprintin g o f th e origina l lx>o k publishe d in 1949 ; however, ther e hav e bee n som e correction s mad e i n th e tex t an d a lis t of errata has been added at th e end o f the book . I n additio n the NOTES whic h follow thi s Introductio n hav e bee n adde d t o thi s edition . Fo r callin g error s to my attention, a s well as for assistanc e wit h th e Notes , I a m indebte d t o bot h colleagues and forme r students .

NOTES T O TH E 196 3 EDITIO N

Page 193 ; 7.2 Theorem . Th e "(P , Q) n+i" conditio n ma y b e replace d b y th e weaker condition "(P, Q, ^)„+i" define d on page 327. Fo r a much simpler proof of thi s theore m se e Theore m VI. 2 o n pag e 22 7 o f m y pape r A certain class of topological properties, Bull . Amer . Math . Soc , vol . 6 6 (1960) , pp . 205-239 .

Chapters VII, VIII. Materia l i n these chapters may b e greatly simplifie d b y the us e o f exac t sequences ; for example , Theore m 2.19 , p. 20 8 (Mayer-Vietori s sequence); Theorems 3.9 , 3.10 , p . 215 ; Theorem 9. 1 o n p . 24 1 (Mayer-Vi e toris sequence); and Lemma s 6.2 , 6. 3 on p . 262 . Shea f theor y ha s bee n applie d b y F. A . Raymond , A . Bore l an d other s t o simplifyin g an d extendin g dualitie s t o other coefficien t domains ; see, e.g., A . Borel , The Poincare duality in generalized manifolds, Mich . Math . Jour. , vol . 4 (1957) , pp . 227-239 ; F . A . Raymond , Poincare duality in homology manifolds, Dissertation , Universit y o f Michigan , 1958; A. Borel and J. C. Moore, Homology theory for locally compact spaces, Mich. Math. Jour., vol . 7 (1960) , pp. 137-159 ; and A . Borel , "Seminar o n Transforma - tion Groups/ ' Princeton , N . J. , Annal s o f Math . Studies , No . 46 , 1960 . I n connection wit h Theore m 9.1 , p. 269, attention shoul d b e called t o K . Sitnikov , The duality law for non-closed sets, Doklady Akad . Nau k SSSR , vol . 8 1 (1951) , pp. 359-362; and i n th e sam e connection , bu t wit h referenc e als o t o dualitie s i n general, see Frank Raymond, Local cohomology groups with closed supports, Math . Zeit., vol. 7 6 (1961) , pp. 31-41.

Page 257. Afte r th e proo f o f 5. 8 Lemma , insert : "As a consequenc e o f Theore m V 18.31 , Theore m 1.1 , and Lemma s 5. 6 an d 5.8, we can sho w

5.8a LEMMA . With P and Q as before, n r Hr+l(S:Q,0;P,0) = h - ~\S: Q, P). (This Lemm a i s needed i n 7.2 , for instance) " xi Xll NOTES

Page 316 ; 1. 1 Theorem . Thi s theore m i s vali d fo r an y orientabl e n-gc m (See my paper A certain class of topological properties, loc. cit., especially Theore m II.5 thereo f an d th e "Remark " followin g it. ) A similar observatio n hold s wit h regard t o the followin g item s i n Chapter XI : Page 319 ; 1. 4 Theore m an d 2. 2 Lemm a Page 319 ; 1. 5 Theore m Page 320 ; 2.1 Theore m Page 321; 2.3 Theore m Page 325 ; 2.19 Theore m (althoug h Theore m V. l o f th e pape r cite d abov e is more general) Page 326 ; 2.20 Corollar y Page 326; 2.21 Corollar y (thi s hold s fo r an y orientabl e close d locall y euclidean n-manifol d S suc h that Pi(S) = 0 , and M nee d b e only 0-lc an d have property (P , Q, ^)n~2. Se e Corollar y V.l o f th e pape r cited above ) Page 326 ; 2.22 Theore m an d Corollarie s (se e Theore m V. 2 o f th e pape r cited above ) Page 329 ; 3.5 Theore m (se e Theorem II.4 o f th e paper cited above ) Page 339 ; 5.12 Theore m (vali d fo r D an y domai n suc h tha t p n~l(D) i s finite, i n an orientable n-gcm ; a lik e remar k hold s fo r Corollar y 5.13 ) Page 340; 5.15 Theore m Page 340; 5.16 Theore m Page 343; 5.26 Theore m Page 344; all Corollarie s 5.27, 5.28 , 5.2 9 an d 5.3 1 Page 345; 6.5 Theorem .

Pages 327 , 328 ; replac e 3. 3 Theore m an d 3. 4 Theorem , respectively , b y Theorems II. 1 and II.2 o f the paper cited above.

Page 366 ; 2.14 Theorem. Th e weake r conditio n "(P , Q, ~ ) r" ma y b e sub- stituted fo r the lc r" condition i n the hypothesis .

Page 381; Problem 1.1 . Solve d affirmatively b y Mary Ellen Estill, A primitive dispersion set of the plane, Duk e Math . Jour. , vol . 1 9 (1952) , pp . 323-328 . Page 381; Problems 1. 2 and 2.3. Se e M. Lubanski , An example of an absolute neighborhood retract, which is the common boundary of three regions in the 3- dimensional euclidean space, Fund. Math., vol. 40 (1953), pp. 29-38. (Notic e also footnote 2) , ibid. , concernin g a n unpublished resul t o f Grub a in 1937. )

Page 381 ; Problem 2.1 . Solve d affirmativel y fo r th e cas e wher e ther e exist s a well-ordere d (b y inclusion ) basis fo r th e ope n neighborhood s o f B. Se e m y paper Some consequences of a method of proof of J. H. C. Whitehead, Mich. Math. Jour., vol. 4 (1957) , pp. 27-31. NOTES xiii

Page 382; Problems 3.1 and 3.2. Fo r solutions for certain types of "homolog y n-manifolds" over th e real s mo d 1 or a principa l idea l ring, see , respectively , C. T . Yang , Transformation groups on a homological manifold. Trans. Amer . Math. Soc, vol . 8 7 (1958) , pp . 261-283; and F . A . Raymond , Poincart duality in homology manifolds, Dissertation , Universit y o f Michigan , 1958 . Page 382; Problem 2.4 . Solved affirmativel y b y R . D . Johnson , A note on uniformly locally connected spaces, Duke Math. J., vol. 30 (1963), pp. 413-416.

Page 383; Problem 4.6. Solve d affirmatively b y R. H. Bing, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Annals o f Math. , vol. 56 (1952) , pp. 354-362.

ADDITIONAL NOTE S FO R TH E 197 9 PRINTIN G Page 382 , Proble m 2.4 . Solve d affirmativel y b y R . D . Johnson , A note on uniformly locally connected open sets, Duke Math. J. 30 (1963), 413-416. Page 382 , Proble m 4.1 . Solve d affirmativel y b y G . E . Bredon , Wilder manifolds are locally orientable, Proc . Nat . Acad . Sci . U.S.A . 6 3 (1969) , 1079-1081. Page 383, Proble m 4.3. Se e M . E . Rudin an d P . Zenor, A perfectly normal nonmetrizable manifold, Houston J. Math. 2 (1976), 129-134. Page 383, Proble m 4.4 . Se e th e references t o F . Raymond , A . Borel an d J. C. Moore , i n th e comments o n Chapter s VII , VII I i n th e Notes t o th e 196 3 Edition, p . xi ; als o G . E . Bredon , Sheaf Theory, McGraw-Hill , Ne w York , 1967.

For furthe r informatio n concernin g th e histor y o f generalize d manifold s since th e publicatio n o f th e presen t work , th e reade r ca n b e referre d t o th e following sources: F. Raymond, R. L. Wilder ys work on generalized manifolds-an appreciation, in Geometri c an d Algebrai c Topology , ed . K . Millett , Proceeding s o f th e Santa Barbara Conference, Lectur e Notes in Math., vol. 664, Springer-Verlag, New York , 1978 . Se e especiall y th e tex t an d Reference s a t the latter part o f the article. R. L . Wilder , The Mathematical Work of R. L. Moore: Its Background, Nature and Influence, Parts Hl b an d Hid , i n R . L . Moore , eds . R . H . Bin g and L . E. Whyburn, Univ . o f Texas Press , Austin, Texas, 1979 . This page intentionally left blank BIBLIOGRAPHY

Only book s tha t ar e o f specia l significanc e fo r th e text an d papers tha t ar e specifically referre d t o therein ar e included i n this bibliography. Fo r papers o n manifolds o f the classical type, not referred to herein, the reader is referred to the bibliographies i n the Colloquiu m volume s o f Lefschet z an d the book o f Seifer t and Threlfal l cite d below . Fo r additional citation s i n the general literatur e o f Topology, reference may be made to the works of Lefschetz and Seifert-Threlfal l as wel l as to the bibliographies i n the Colloquium volume s o f R. L. Moore and G. T. Whyburn cite d below .

BOOKS

ALEXANDROFF, P . an d HOPF , H . [A-H] Topologie, Berlin , Springer , 193 5 (Die Grundlehren de r mathematischen Wissen - schaften, bd . 45). HAUSDORFF, F . [H] Grundziige der Mengenlehre, Leipzig , vo n Weit, 1914. [Hi] Mengenlehre, Berlin , d e Gruyter, 1927 , 1935 . HILBERT, D . an d COHN-VOSSEN , S . [H-C] Anschauliche Geometrie, Berlin , Springer , 1932 ; in reprint, N . Y., Dover, 1944 . HUREWICZ, W . an d WALLMAN , H . [H-W] Dimension Theory, Princeto n Univ . Pr. , 194 1 (Princeto n Mathematica l Series , no. 4). KEREKJARTO, B . VON . [K] Vorlesungen ihber Topologie, I , Fldchentopologie, Berlin , Springer , 1923. KURATOWSKI, K [K] Topologie I , Warsaw, 1933 ; 2d ed., 1948. LEFSCHETZ, S . [L] Algebraic Topology, Ne w York, 194 2 (American Mathematical Societ y Colloquiu m Publications, vol . 27). [Li] Topics in Topology, Princeto n Univ . Pr . 194 2 (Annals o f Mathematic s Studies , no. 10).

[L2] Topology, Ne w York , 193 0 (America n Mathematica l Societ y Colloquiu m Pub - lications, vol. 12). MENGER, K . [Me] Kurventheorie, Berlin , Teubner , 1932 . [Mei] Dimensionstheorie, Berlin , Teubner , 1928. MOORE, R . L . [Mo] Foundations of Point Set Theory, Ne w York , 193 2 (America n Mathematica l Society Colloquiu m Publications , vol . 13). NEWMAN, M . H . A . [N] Elements of the Topology of Plane Sets of Points, Cambridg e Univ. Pr. , 1939. PONTRJAGIN, L . [P] Topological Groups, Princeto n Univ . Pr. , 193 9 (Princeto n Mathematica l Series , no. 2). REIDEMEISTER, K . [R] Topologie der Polyeder, Leipzig , Akademisch e Verlagsgesellschaft , 1938 .

385 386 TOPOLOGY O F MANIFOLD S

SCHOENFLIES, A . [S] Die Entwickelung der Lehre von den Punktmannigfaltigkeiten, II , Leipzig, Teubner , 1908 (Jahresbcricht der Deutschen Mathematiker-Vereinigung , Erganzungsbande , bd. II). SEIFERT, H . an d THRELFALL , W . [S-T] Lehrbuch der Topologie, Leipzig , Teubner, 1934 . SlERPINSKI, W . [S] Introduction to General Topology , Univ . o f Toronto Pr. , 1934. [Si] Lecons sur les Nombres Transftnis, Paris , Gauthier-Villars, 1928. TUCKER, A . W . [T] Elementary Topology, mimeographe d note s b y F . W . Burto n an d J . H . Lewis , Princeton University , 1935-36 . VEBLEN, O . [V] Analysis Situs, 2 d ed. , Ne w York , 193 1 (America n Mathematica l Societ y Col - loquium Publications , vol . 5, part 2). WHYBURN, G . T .

[Wh] Analytic Topology t Ne w York, 194 2 (American Mathematica l Societ y Colloquiu m Publications, vol . 28). WILDER, R . L . an d AYRES , W . L. , editor s [UM] Lectures in Topology, Th e Universit y o f Michiga n Conferenc e o f 1940 , Univ. o f Mich. Pr., 1941.

PAPERS, ETC.

ALEXANDER, J . W .

[a] A proof and extension of the Jordan-Brouwer separation theorem f Transaction s o f the America n Mathematica l Society , vol . 23 (1922), pp. 333-349. [b] Combinatorial analysis situs, I , Transaction s o f th e America n Mathematica l Society, vol. 28 (1926), pp. 301-329. [c] An example of a simply connected surface bounding a region which is not simply connected, Proceeding s o f the National Academ y o f Sciences , vol . 1 0 (1924) , pp. 8-10. [d] Note on Pontrjagin's topological theorem of duality, Proceeding s o f th e Nationa l Academy o f Sciences, vol. 21 (1935), pp. 222-225. [e] Manifolds, Encyclopaedi a Britannica , 14t h ed., vol. 14, pp. 804-805. ALEXANDROFF, P . [ai] fiber stetige Abbildungen kompakter Raume, Proceeding s o f the Amsterdam Acad - emy, vol . 28 (1925), pp. 997-999.

[a2] t)ber stetige Abbildungen kompakter Raume, Mathematisch e Annalen , vol . 9 6 (1926-27), pp. 555-571. [b] Zur Begrundung der n-dimensionalen mengentheoretische Topologie, Mathematisch e Annalen, vol . 94 (1925), pp. 296-308. [c] Untersuchungen uber Gestalt und Lage abgeschlossener Mengen beliebiger Dimension, Annals o f Mathematics (2) , vol. 30 (1928-29) , pp. 101-187. [d] (wit h Urysohn, P.) Mtmoire sur les espaces topologiques compacts, Verhandelinge n der Koninklijke Akademi e van Wetenschappen te Amsterdam, Afdeelin g Natuur - kunde (Erst e Sectie) , Deel XIV, No. 1 (1929). [e] Zur Homologie-Theorie der Kompakten, Compositi o Mathematica , vol . 4 (1937) , pp. 256-270 . [f] On local properties of closed sets, Annals of Mathematics (2) , vol. 36 (1935), pp. 1-35. [g] Einfachste Grundbegriffe der Topologie, Berlin , Springer , 1932. [h] Une g&n&ralisation nouvelle du thiorhme de Phragm&n-Brouwer, Compte s Rendus de TAcademie de s Sciences, Paris, vol. 184 (1927), pp. 575-577. BIBLIOGRAPHY 387

[i] (wit h Pontrjagin , L. ) Les varUtis a n-dimensions g&nbralise'es, Compte s Rendu s de TAcademi e de s Sciences , Paris, vol. 20 2 (1936) , pp. 1327-29 . [j] General combinatorial topology, Transaction s o f th e America n Mathematica l Society, vol. 49 (1941) , pp. 41-105. [k] On homological situation properties of complexes and closed sets, Transactions o f th e American Mathematica l Society , vol . 5 4 (1943) , pp. 286-339 . BEGLE, E . [a] Locally connected spaces and generalized manifolds, America n Journa l o f Mathe - matics, vol. 64 (1940) , pp. 553-574 . [b] Duality theorems for generalized manifolds, America n Journa l o f Mathematics , vol. 6 7 (1945) , pp. 59-7 0 [c] A note on local connectivity, Bulleti n o f the American Mathematical Society, vol. 54 (1948), pp. 147-148 . BING, R . H . [a] The Kline sphere characterization problem, Bulleti n o f the America n Mathematica l Society, vol . 52 (1946) , pp. 644-653. BROUWER, L . E . J . [a] Zur analysis situs, Mathematisch e Annalen , vol . 68 (1910) , pp. 422-434 . [b] Beweis des Invarianz der geschlossenen Kurve, Mathematisch e Annalen , vol . 7 2 (1912), pp. 422-425 . [c] Beweis des Jordanschen Satzes fur den n-dimensionalen Raum, Mathematisch e Annalen, vol . 7 1 (1912) , pp. 314-319 . [d] fiber Jordanschen Mannigfaltigkeiten, Mathematisch e Annalen , vol . 7 1 (1912) , pp. 320-327 . [e] Beweis des Jordanschen Kurvensatzes, Mathematisch e Annalen , vol . 6 9 (1910) , pp. 169-175 . CECH, E . [a] ThSorie ge'nirale de Vhomologie dans un espace quelconque, Fundament a Mathema - ticae, vol . 1 9 (1932) , pp. 149-183 . [b] Theorie generate des varie'te's et de leurs th&oremes de dualiU, Annal s o f Mathematic s (2), vol. 34 (1933) , pp. 621-730 . [c] Sur la dimension des espaces parfaitement normaux, Bulleti n internationa l d e TAcademie de s Science s de Boheme, 1932 , pp. 1-18 . [d] Sur la connexite' local d'ordre supirieur, Compositi o Mathematica , vol . 2 (1935) , pp. 1-25 . [e] On general manifolds, Proceeding s o f th e Nationa l Academ y o f Sciences , vol . 2 2 (1936), pp. 110-111 . [f] Sur les nombres de Betti locaux, Annal s o f Mathematic s (2) , vol . 3 5 (1934) , pp. 678-701 . [g] On pseudomanifolds (mimeographed) , Princeton , N . J. , Th e Institut e fo r Ad - vanced Study , 1935 . [h] Sur la decomposition d'une pseudovarie'te' par un sousensemble ferme", Compte s Ilendus de VAcademic de s Sciences, vol. 19 8 (1934), pp. 1342-1345 . CHITTENDEN, E . W . [a] On the metrization problem and related problems in the theory of abstract sets, Bulleti n of th e America n Mathematica l Society , vol . 33 (1927) , pp. 13-34 . EILENBERG, S . [a] Singular homology theory, Annal s o f Mathematics , vol . 45 (1944) , pp. 407-447 . FRANKL, F . [a] Topologische Beziehungen in sich kompakter Teilmengen euklidischer Rdume zu ihren Komplementen sowie Anwendung auf die Prim-Enden-Theorie, Akademi e de r Wissenschaften, Wien , Math.-Naturw . Kl. , Sitzungsberichte , Abt . 2A , vol . 13 6 (1927), pp. 689-699 . 388 TOPOLOGY OF MANIFOLDS

GAWEHN, I . [a] Uber unberandete zweidimensionale Mannigfaltigkeiten, Mathematisch e Annalen , vol. 98 (1927) , pp. 321-354. GEHMAN, H . M . [a] Concerning irreducibly connected sets and irreducible continua, Proceeding s o f th e National Academ y o f Sciences , vol. 1 2 (1926) , pp. 544-547 . [b] Irreducible continuous curves, America n Journa l o f Mathematics , vol . 4 9 (1927) , pp. 189-196 . [c] Concerning irreducible continua, Proceeding s o f the National Academy o f Sciences , vol. 1 4 (1928) , pp. 43P-435. HAHN, H . [a] Uber die Komponenten offener Mengen, Fundament a Mathematica , vol . 2 (1921) , pp. 189-192 . JORDAN, C . [a] Cours d'Analyse, 2 ed., Paris , Gauthier-Villars , 1893 , vol. 1 . VAN KAMPEN , E . R . [a] On some characterizations of 2-dimensional manifolds, Duk e Mathematica l Journal , vol. 1 (1935) , pp. 74-93 . [b] Die kombinatorische Topologie und die Dualitdtssatze, Th e Hagu e (1929) . (Leyde n thesis). KAPLAN, S . [a] Homologies in metric separable spaces, Dissertation , Universit y o f Michigan , An n Arbor, 1942 . [b] Homology properties of arbitrary subsets of euclidean spaces, Transaction s o f th e American Mathematica l Society , vol . 6 2 (1947) , pp. 248-271 . KNASTER, B . an d KURATOWSKI , C . [a] Sur les ensembles connexes, Fundament a Mathematicae , vol . 2 (1921) , pp. 206-255. [b] Sur les continus non-bornfe, Fundament a Mathematicae , vol . 5 (1924) , pp. 23-58. KURATOWSKI, C . [a] Une definition topologique de la ligne de Jordan, Fundament a Mathematicae , vol. 1 (1920), pp. 40-43 . [b] Une caracterisation topologique de la surface de la sphere, Fundament a Mathe - maticae, vol . 1 3 (1929) , pp. 307-318 . KLINE, J . R . [a] Closed connected sets which remain connected upon the removal of certain connected subsets, Fundament a Mathematicae , vol . 5 (1924) , pp. 3-10 . [b] Separation theorems and their relation to recent developments in analysis situs, Bulletin o f the America n Mathematica l Society , vol. 34 (1928) , pp. 155-192 . LEBESGUE, H . [a] Sur les correspondances entre les points de deux espaces, Fundamenta Mathematicae , vol. 2 (1921) , pp. 256-285. LEFSCHETZ, S . [a] The residual set of a complex on a manifold and related questions, Proceeding s o f th e National Academy o f Sciences, vol. 1 3 (1927), pp. 614-622, pp. 805-807. [b] Closed point-sets on a manifold, Annal s o f Mathematic s (2) , vol . 2 9 (1928) , pp. 232-254 . [c] On generalized manifolds, America n Journa l o f Mathematics , vol . 5 5 (1933) , pp. 469-504 . [d] Chain-deformations in topology, Duke Mathematical Journal, vol. 1 (1935), pp. 1-18 . LENNES, N . J . [a] Bulleti n o f th e America n Mathematica l Society , vol . 1 2 (1906) , p . 284 , abstrac t no. 10 . [b] Curves in non-metrical analysis situs with applications in the calculus of variations, American Journa l o f Mathematics , vol . 33 (1911) , pp. 287-326 . BIBLIOGRAPHY 389

MAZURKIEWICZ, S . [a] Sur un probVeme de M. Knaster, Fundament a Mathematicae , vol . 1 3 (1929) , pp. 146-150 . MENGER, K . [a] Grundziige einer Theorie der Kurven, Mathematisch e Annalen , vol . 9 5 (1925) , pp. 277-306 . MOORE, R . L . [a] On the most general class L of Frechet in which the Heine-Borel-Lebesgue theorem holds true, Proceedings o f the National Academy o f Sciences, vol. 5 (1919), pp. 206- 210. [b] Concerning simple continuous curves, Transaction s o f the American Mathematica l Society, vol. 21 (1920), pp. 313-320. [c] Concerning continuous curves in the plane, Mathematisch e Zeitschrift , vol . 1 5 (1922), pp. 254-260. [d] Concerning connectedness im kleinen and a related property, Fundament a Mathe - maticae, vol. 3 (1922) , pp. 232-237. [e] Report on continuous curves from the viewpoint of analysis situs, Bulleti n o f th e American Mathematica l Society , vol . 29 (1923) , pp. 289-302. [f] A characterization of Jordan regions by properties having no reference to their bound- aries, Proceeding s o f the National Academy o f Sciences, vol. 4 (1918), pp. 364-370. [g] On the relation of a continuous curve to its complementary domains in space of three dimensions, Proceeding s o f th e Nationa l Academ y o f Sciences , vol . 8 (1922) , pp. 33-38 . POINCARE, H . [a] Analysis situs, Journa l d e l'Ecole Polytechnique , (2) , vol. 1 (1895), pp. 1-121 . PONTRJAGIN, L . [a] Uber den algebraischen Inhalt topologischer Dualitdtssdtze, Mathematisch e Annalen , vol. 10 5 (1931), pp. 165-205. [b] The general topological theorem of duality for closed sets, Annal s o f Mathematics , (2), vol. 35 (1934), pp. 904-914. [c] Zum Alexanderschen Dualitdtssatz, Gottinge r Nachrichten , Math.-Phys . KL, 1927, pp. 315-322 . [d] Zum Alexander schen Dualitdtssatz, Zweit e Mitteilung , Gottinge r Nachrichten , Math.-Phys. KL, 1927, pp. 446-456. SCHOENFLIES, A . [a] Bemerkungen zu dem vorstehenden Aufsatz des Herrn L. E. J. Brouwer, Mathe - matische Annalen , vol . 68 (1910) , pp. 435-444. SlERPINSKI, W . [a] Sur une condition pour qu'un continu soit une courbe jordanienne, Fundament a Mathematicae, vol . 1 (1920), pp. 44-60. STEENROD, N . E . [a] Universal homology groups, America n Journa l o f Mathematics , vol . 5 8 (1936) , pp. 661-701 . SWINGLE, P . M . [a] An unnecessary condition in two theorems of analysis situs, Bulleti n o f the America n Mathematical Society , vol . 34 (1928), pp. 607r618. TlETZE, H . [a] Uber die topologischen Invarianten mehrdimensionalen Mannigfaltigkeiten, Monat - shefte fu r Mathemati k un d Physik, vol . 1 9 (1908), pp. 1-118. TORHORST, M . [a] Uber den Rand der einfach zusammenhdngenden ebenen Gebiete, Mathematisch e Zeitschrift, vol . 9 (1921) , pp. 44-65. 390 TOPOLOGY OF MANIFOLDS

URYSOHN, P . [a] Uber die Metrization der kompakten topologischen Rdume, Mathematisch e Annalen , vol. 9 2 (1924) , pp. 275-293. [b] Uber die Mdchtigkeit der zusammenhdngenden Mengen, Mathematisch e Annalen , vol. 9 4 (1925) , pp. 262-295. [c] Zum Metrizationsproblem, Mathematisch e Annalen , vol . 9 4 (1925) , pp . 309-315 . [d] Uber im kleinen zusammenhdngenden Kontinua, Mathematisch e Annalen , vol . 9 8 (1927), pp. 296-308. [d] Me'moire sur les multiplicity Cantoriennes, Fundament a Mathematicae , vol . 7 (1925), pp. 30-137 . '

[e2] Memoire sur les multiplicity Cantoriennes, Fundament a Mathematicae , vol . 8 (1926), pp. 225-356 . VAUGHAN, H . E . [a] On local Betti numbers, Duk e Mathematica l Journal , vol . 2 (1936) , pp. 117-137 . VEBLEN, O . an d ALEXANDER , J . W . [a] Manifolds of N dimensions, Annal s of Mathematics (2) , vol. 1 4 (1913), pp. 163-178. VIETORIS, L . [a] Uber den hoheren Zusammenhang kompakter Rdume und eine Klasse von zusammen- hangstreuen Abbildungen, Mathematisch e Annalen , vol . 97 (1927) , pp. 454-472 . WHITNEY, H . [a] On products in a complex, Annal s o f Mathematic s (2) , vol. 39 (1938) , pp. 397-432. [b] A characterization of the closed 2-cell, Transactions o f th e America n Mathematica l Society, vol . 3 5 (1933) , pp. 261-273. WHYBURN, G . T . [a] Semi-locally connected sets, America n Journa l o f Mathematics , vol . 6 1 (1939) , pp. 733-749 . [b] Concerning accessibility in the plane and regular accessibility in n dimensions, Bulletin o f the America n Mathematica l Society , vol . 3 4 (1928) , pp. 504-510 . [d] Concerning regular and connected point sets, Bulletin o f the American Mathematica l Society, vol . 33 (1927) , pp. 685-689 . [e] A generalized notion of accessibility, Fundament a Mathematicae , vol . 1 4 (1929) , pp. 311-326 . [f] On the cyclic connectivity theorem, Bulleti n o f the America n Mathematica l Society , vol. 3 7 (1931) , pp. 429-433. [g] Concerning continuous images of the interval, America n Journa l o f Mathematics , vol. 5 3 (1931) , pp. 670-674 . [h] On the construction of simple arcs, America n Journa l o f Mathematics , vol . 5 4 (1932), pp. 518-524 . [i] Continuous curves without local separating points, America n Journa l o f Mathe - matics, vol. 53 (1931) , pp. 163-166 . WILDER, R . L . [a] Concerning continuous curves, Fundamenta Mathematicae , vol . 7 (1925) , pp. 340 - 377. [b] On a certain type of connected set which cuts the plane, Proceeding s o f th e Inter - national Mathematica l Congress , Toronto, 1924 , pp. 423-437 . [c] A converse of the J ordan-Brouwer separation theorem in three dimensions, Transac - tions o f the America n Mathematica l Society , vol . 32 (1930) , pp. 233-240 . [d] Point sets in three and higher dimensions and their investigation by means of a unified analysis situs, Bulleti n o f th e America n Mathematica l Society , vol . 3 8 (1932) , pp. 649-692 . [e] A theorem on connected point sets which are connected im kleinen, Bulleti n o f th e American Mathematica l Society , vol . 3 2 (1926) , pp. 338-340 . BIBLIOGRAPHY 391

[f] A point set which has no true quasi-components, and which becomes connected upon the addition of a single point, Bulleti n o f th e America n Mathematica l Society , vol. 3 3 (1927) , pp. 423-427 . [g] On connected and regular point sets, Bulletin o f the American Mathematical Society , vol. 3 4 (1928) , pp. 649-655 . [h] A characterization of continuous curves by a property of their open subsets, Funda - menta Mathematicae , vol . 1 1 (1928), pp. 127-131 .

[i] On the linking of Jordan continua in E n by (n — 2)-cycles % Annals o f Mathematic s (2), vol. 34 (1933) , pp. 441-449 . [j] Concerning simple continuous curves and related point sets, America n Journa l o f Mathematics, vol . 53 (1931) , pp. 39-55. [k] On the properties of domains and their boundaries, Mathematisch e Annalen , vol . 109 (1933) , pp. 273-306 . [m] Concerning irreducibly connected sets and irreducible regular connexes, America n Journal o f Mathematics , vol . 56 (1934) , pp. 547-557 . [n] Generalized closed manifolds in n-space, Annal s o f Mathematics (2) , vol. 35 (1934) , pp. 876-903 . [o] On locally connected spaces, Duke Mathematical Journal, vol. 1 (1935), pp. 543-555. [p] Sets which satisfy certain avoidability conditions, Casopi s pro Pestovanl Matematik y a Fysiky , vol . 6 7 (1938) , pp. 185-198 .

[q] Property S n , American Journal o f Mathematics, vol. 61 (1939), pp. 823-832. [s] The sphere in topology, America n Mathematica l Societ y Semicentennia l Publica - tions, Ne w York , 1938 , vol. 2, pp. 136-184 .

[t] A characterization of manifold boundaries in E n dependent only on lower dimensional connectivities of the complement, Bulleti n o f th e America n Mathematica l Society , vol. 42 (1936) , pp. 436-441 .

[Ai]-[Ai3] Thes e ar e abstract s o f unpublishe d paper s i n th e Bulleti n o f th e America n Mathematical Societ y wit h volume , pag e an d abstrac t no . a s follows : Ai—3 5

(1929), 194 , 9 ; A 2—43 (1937) , 335 , 272 ; A3—5 2 (1946) , 446 , 219; A 4—52 (1946) , 445, 217; A5—46 (1940) , 57, 134; A6—52 (1946) , 445, 216; A7—46 (1940) , 436, 349; A8—52 (1946) , 446 , 218 ; A 9—47 (1941) , 58 , 114 ; A 10—47 (1941) , 58, 113 ; An — 47 (1941) , 58, 112 ; A12—42 (1936) , 496 , 308; A13—53 (1947) , 507, 287. YOUNG, G . S. , JR . [a] A characterization of 2-manifolds, Duk e Mathematica l Journal , vol . 1 4 (1947) , pp. 979-990 . ZIPPIN, L . [a] On continuous curves and the Jordan Curve Theorem, America n Journa l o f Mathe - matics, vol. 52 (1930) , pp. 331-350 . [b] A characterization of the closed 2-cell, America n Journa l o f Mathematics , vol . 5 5 (1933), pp. 207-217 . This page intentionally left blank INDEX O F SYMBOL S (The symbol "f " refer s to a footnote; th e symbol "ff " t o the following pag e or pages.)

C, D, 1 < , 178 , 222 ~ , 55 , 124 C, 3, 6 > , 129 , 17 1 =,16 9 ! } , Ml, 2 » , 13 3 — , 15 1 ff H, U, 1 () , 2 7 ^ , 153 , 158, 247 ff U,U„,n. C\n,2 & , 2 A , 131 , 153, 203 a, 5 3 ff , 12 1 v , 2 e , i 5, 14 9 -» , 7 || , 5 3 «( ) , 5 3 f • , 153 , 162 , 164 , 24 7 f f | | || , 5 5 3), 12 8 it , 12 6 M, 5 p( ) , 4 r , 17 7 T* , 25 2

C.,,82 ff(A, B)M , 46 8\ S n, 31 £>„(£>, S ; L, P), 25 2 H*(A, B)M, 4 6 S., 23 4 ff S„(S;L, P), 25 2 Ki, 5 6 f, 12 2 S- , 238 £\ 2 1 (L", Q"), 15 9 S(x, e), 4 Ml,. , 316 FT, E", 3 1 St(£T), 12 0 (P, Q) , 193 F( ),se e Boundar y n St(M, U), 133 (P, Q)" , 237 F(x, e), see Boundary St(23, U) , 133 (p,Q,~)r,(p,Q,~y, F. , 16 8 327 {V,,Px,}, 147 , 151

G, , 16 8 S, 106 , 339 WSr , WSl , 342

UPiM, 13 1 U*(U; M, P), 18 2 23*(U, 33 ; Jlf, P) , 18 2 U A Af , 13 1 Ui(U;P), 18 3 KQ1, 23 ; P), 18 3 uns, 12 9 U„*(U; P), 18 4 «*(U, 93 ; P), 18 4 Ui(U; M, P), 18 2 SKfll, 33 ; .¥, P), 18 2 393 394 TOPOLOGY O F MANIFOLD S

-cole, 18 9 ff lcn, 17 8 lirh, 55 , 121 , 126 , 13 6 coJcm, eole r , 19 0 lej , 19 7 lp,3 -coulc, 190 , 292 ff -LC, 28 7 sip, 7 3 -Culc, 19 0 leg, 4 0 ucos, 16 9 Culcn, 17 8 les, 14 4 ulc, 65 , 10 9 fcos, 12 9 lew, 4 0 -ule, 65 , 292, 29 6 -gem, 24 4 lirru , 15 1 ulc*, ulcl , 29 9 -gm, 24 4 linv , 14 7 -wulc, 34 4 le, 4 0 Km sup, 10 2 wulcj , 344 -lc, 17 6 lircoh, 16 5

CHAIN AN D HOMOLOG Y GROUP S Chain group s are designate d b y the letter "C " followed b y suitabl e symbols ; for example , "Cr(K)." Cycl e groups and group s o f bounding cycle s are designated similarl y by use o f the letters "Z" and "B'\ respectively . Consequentl y i n the table belo w only the symbols for th e homology groups are given; to obtain the corresponding chain, cycle or bounding cycle groups (where they exist), replace "H" b y "C," "Z," or "5." Thu s the cycl e group corresponding to r r r "H*(K)" i s "Z (K):' Henc e to look up a group, as "Z (S; M, L; G)t" instea d look up "H (S; M, L; G)" in the index below; the "Z" group desired wil l be found define d o n the page cited. Betti number s are generall y designate d b y "p." Thus , to loo k up the meanin g o f "p r(S: M, L; A, B)," tur n to the page designated for "H r(S: M, L ; A, B)." Simila r remarks hold fo r the cohomology case. However , some Betti and co-Betti numbers are listed below, particularly where special definitions ar e required or a letter differen t fro m "p" (suc h as "g") i s used. Individual cycle s and cocycle s ar e variousl y denote d i n the tex t b y th e letter s "z, " "Z," "y" an d "r" wit h appropriat e indices . Ope n poin t set s ar e denote d belo w b y "P, " "Q," "U," "V;" close d point sets by "A," "£, " "J," "M," "L;" a single point by "*." Also , both below and in the text, the letter "2C " denotes a complex, li S" a space, -"S^ a n algebrai c field , and "G u a n abelian grou p

H'(K), 54 H"(M; J, 0), 211 h\U, V), 235 Hr(U), 58 Hn(S;M,L;5), 136 h"((P), 235 Hr(K; G), 123 ET(S), H"(S; 5), 136 h'(S), 246 Hl(K; G), 124 Hr(S: M, L; A, B), 166 hT(S: Q; P), 256 H\S; G), 130 H',(S), 188 &(S), 248 H"(M, L; G, U), 131 Hr,(S; S, M), 261 &(S), 258 n H (S; M, L; G), 132 H'fa{M), 262 H"(x;P,Q), 190

H"(M, L; G), 133 H'P(M, ff), 172 CT(M; J, 0), 211 H"(M, L), 133 K{P; ff),18 5 INDEX O F SYMBOLS

Hn(K; G), 15 0 HJLP, Q), 152 hr(S), 248

Hn{P,Q;S, U) , 15 0 ff Hn(S), 152 £,0S), 247

Hn(S;P, Q;JF) , 152 ffr(5: P, Q; U, V), 166 #(S), 260

Hn(S;P, Q) , 152 Hn(x;P,Q), 191 p'(tf, 2), 55

Abstract space , 11 , 1 3 Basis, homology ; se e Base , homolog y Accessibility, 1 3 Basis, countabl e (o f ope n sets) , 7 0 arcwise, 6 6 Betti group ; se e Homology grou p by close d an d connecte d sets , 11 0 Betti number, 1 2 by continua , 11 0 around a point, 29 1 from al l sides, 11 6 co-, 16 3 local r-, 36 2 local, loca l co- ; se e Loca l connectivit y r-, 36 2 numbers regular, 116 , 35 3 mod 2 , o f a euclidean complex , 5 5 regular r-, 35 4 of a space , 16 3 regular r- , rel . G r, 35 9 of K ove r G, 12 4 semi-r-, 36 8 of open subset s o f S n, 5 8 theorems, 11 0 ff, 11 6 ff, XII Boundary uniform semi-r- , 37 7 of a set o f points, 1 6 ff. Th e boundary o f a Acyclic, 8 5 point se t M i s denote d b y th e symbo l Addition theorems , 60 , 6 4 ff, 241 ff F(M); th e boundary of an ^neighborhoo d Alexander additio n theorem , 6 0 of a point x in a metric space by F(x f e) . Alexander dualit y theorem , 14 , 6 3 of a chain , 54 , 12 2 for a n n-gem , 26 3 ff Boundary cell , 5 2 Algebraic topology , 12 , 52 Boundary chain , 12 1 ff Analysis situs, 1 0 Boundary operator , 12 1 Annihilator, 15 9 Boundary point , 16-1 7 Arc, 2 7 Brouwer property , 4 7 characterizations of , 3 0 ff, 43 ff possession o f b y S n, 6 0 open, 2 7 Arcwise connected , 8 1 Canonical pai r o f neighborhoods , 19 2 locally, 8 1 Canonical sequenc e o f refinements , 14 5 through a space , 8 1 Cantor-connected ( = C-connected) , 33 8 Arcwise connectedness o f domains o f a Pean o Cantor produc t theorem , 2 4 space, 8 1 Cantor ternar y set , 7 4 Augment, 12 1 Cap product , 15 3 Avoidable point , 22 9 of compact cycles and infinite cocycles , etc., almost r- , 23 1 247 ff in th e relativ e sense , r- , 28 1 of C-eycle s and cocycles , 15 7 ff locally r- , 21 8 special, 15 4 r-, 21 8 Carrier Avoidable se t approximate, 20 5 almost completel y r- , 23 8 minimal, 20 5 almost locall y r- , 23 8 of a C-cycle , 20 4 of a compac t cocycle , 24 8 Barrier, 21 6 of a compac t cycle , 24 6 ff Base, homology , 13 6 of a homology , 20 4 Base o f cycles relative to homology = a set of Cauchy sequence , 4 cycles consistin g o f on e cycl e fro m eac h C-cycle = Cec h cycl e element o f a homolog y base . Bas e o f Cech cocyclcs relative to cohomology i s define d absolute—cycle, 13 2 similarly. cycles, 130 , 14 8 Basic set , abou t whic h a spac e i s irreducibl y cycle mo d L o n M , 13 2 connected, 2 3 unrestricted—homology theory , 16 9

396 INDEX 397

Cell, 11 , 15, 12 0 f Compact, 3 4 characterization o f close d 2- , 92, 24 2 countably, 8 closed, 31 , 55 locally, 2 9 closed 2- , 92 ff locally peripherall y countably , 2 9 n-, 31 , 38, 5 2 Complement, 1 c-equi valence, 1 8 Complete famil y o f coverings , 13 0 Chain, 1 2 Complete space , 4 augmented, 12 3 Completely normal , 5 0 cellular, 5 3 Completely r-avoidable , 22 9 deformation-, 12 8 almost, 23 1 group, 54 , 12 2 f f sets, 23 8 product, 15 3 Complex, 11 , 53 r-, 53 , 54, 12 1 f f associated—of a chain , 5 3 Chain-homo topic, 12 9 augmented, 12 1 Chain-mapping, 12 7 f f cone-, 12 6 Chain-realization, 17 7 deformation-, 12 8 extension of , 17 7 infinite, 12 6 partial, 17 7 n-dimensional, 12 0 Character. A point z o f a Hausdorf f spac e S oriented, 12 1 has countabl e characte r a t x i f som e unrestricted, 12 0 countable collectio n o f ope n subset s o f S Component, 1 8 is equivalen t t o th e se t o f al l ope n set s Condensation, r-dimensional , 35 6 relative t o x. (Th e obviou s generaliza - Connected, 7 tion t o characte r o f typ e a wher e a i s in th e sens e o f Cantor , 33 7 any cardina l number) . space, 16 , 1 9 See 7 3 strongly, 22 7 Closed, 5 Connectedness, relation o f homology to, 14 1 ff Closed cantoria n manifolds , 20 7 Connectivity, 1 1 Closed curve , 13 , 1 4 number, 12 , 15 , 12 4 (se e Betti number ) quasi-, 3 2 number abou t a set, 193 , 240 simple, 3 1 Constituant. Define d exactl y like component Closed Jordan curve , 3 1 (18) excep t tha t tw o point s x an d y ar e Closure o f a poin t set , 5 called c-equivalen t i f ther e exist s a com - Coboundary, 15 0 pact, connecte d se t containin g x an d y. Cochain realization , 252 ; may b e abbreviate d (Compare strongl y connected) . to "co-realization " Continuous curve, 13. Se e Peano continuum . partial, 25 2 Jordan's definition , 6 9 Cocycle, 14 9 f f Continuum = nondegenerate , compac t an d compact, 24 8 connected, 36. (Beginnin g with Chapte r fundamental, 250 , 25 5 IV, a continuu m i s alway s a Hausdorf f infinite, 24 7 space.) of a space , 15 2 Convergent sequenc e o f points , 7 3 Coefficients (o f chains) , 12 , 12 1 f f Coordinate o f a Cech cycle , 13 0 integral, 12 , 12 5 f f Countable. A se t i s calle d countabl e i f i t i s mod 2 , 12 , 52 ff , 12 4 f f empty, finite o r denumerabl e (q.v. ) mod p , 12 , 12 6 Countable bas e o f ope n sets , 7 0 rational, 12 , 12 6 Cover, To , 3 3 Cofinal (directe d systems) , 14 7 Covering, 13 , 12 9 Cohomology, 15 1 neighborhood, 17 1 compact—group, 24 8 regular wit h respec t t o a set , 13 4 group, 14 9 ff , 16 6 theorems, 11 , 35, 106 , 129 , 13 3 ff , 140 , 145, infinite—group, 24 7 169 ff , 173 , 202 Combinatorial topology , 1 1 unrestricted, 16 8 398 INDEX

Cut point , 1 0 Distance function , 4 Cycle, 12 , 54 Domain = ope n connecte d subset o f a space , absolute, 13 2 14 f , 5 2 f approximately o n a set, 17 2 Dot product , 15 3 bounding, 12 , 54, 123 , 13 0 of compact cycles and infinite cocycles, etc., compact, 24 6 247 ff Cech, 130 , 14 8 of Cech cycle s an d cocycles, 16 2 essential, 14 0 Dual bases , 161 , 16 4 ff fundamental, 250 , 25 5 Dual homomorphism , 14 8 ff group, 5 4 Dual pairings , 16 2 infinite, 24 8 orthogonal, 16 2 non-trivial, 142 , 21 6 Duality relative, 13 1 simple local , 27 4 0-, 5 4 Duality theorem s Cyclic, 8 5 between homology an d cohomolog y group s element, 8 2 of a space, 163 , 166 , 247 ff, 256 ff Cyclicly connected , 8 5 between l c and avoidability properties , 340 Cyclic connectivit y theorem , 8 5 between l c and S properties, 32 0 ff between lc and weak S properties, 339,343f f Declinable, r- , 37 1 between l c and wulc properties , 344f f Dedekind Cut Axiom, 29. A simpl y ordere d between S an d avoidabilit y properties , set S i s said to satisfy th e Dedekind Cu t 331 ff Axiom i f for every decomposition S = A between S properties, 31 6 ff, 327 ff U B such that A 5 * 0 7 * B an d A < B, between ul c and coulc , 29 4 either A ha s a last point and/or B ha s a for a complex, 16 1 ff first point . for local Betti numbers , 191 , 193 , 291 ff Deformation-chain, 12 8 for S r i n S n, 6 1 ff Deformation-complex, 12 8 Poineare typ e of , 253 , 25 9 Dense, 8 3 Pontrjagin typ e o f (linking) , 26 6 ff, 30 2 Denumerable. A set is called denumerable i f relation t o separatio n o f spac e b y close d there exist s a (l-l)-correspondenc e be - sets, 21 2 ff, 225 ff tween it s element s an d th e natura l numbers 1 , 2 , • • • , n , • • • . I n th e End poin t terminology o f cardina l numbers , a se t of a n arc , 2 7 is denumerable i f the cardina l number o f of a Peano continuum , 8 2 its element s i s aleph-null . (Se e Count - Equivalent neighborhoo d systems , 3 able.) relative t o a set, 7 2 Diameter o f a point set M = lu b p(x , y), x , y Euclidean plane, top'l characterization of , 28 0 G M. Denote d symbolicall y by 5(M) . Extendible, r- , 22 4 Diameter in the relativ e sense , 28 1 of a chain, 17 8 of a C-cycle , 22 2 Face, 11 , 53, 12 0 of a point set i n a metri c space , 5 3 f Flat, 13 6 of a point set i n genera l space, 10 6 Fundamental parallelopipe d o f Hilbert space , Difference o f sets , 1 71 Dimension Fundamental syste m o f cycles , 145 , 18 6 ff of a chain , 12 1 metric, 21 8 of a covering, 19 5 of infinit e r-cycles , 25 8 of a space, 19 5 ff of infinit e r-cocycles , 26 0 Directed system , 14 7 Disconnect, 1 0 Generalized close d manifold , 24 4 Disjoint, 1 Generalized close d n-cell, 28 7 Distance betwee n tw o poin t sets , 5 8 f Generalized manifol d o f dimensio n n, 24 4 INDEX 399

Generalized manifolds , 1 5 Irreducible lc-connex e about a point set , 4 2 Generalized n-cell, 287 Irreducible membrane, 209 Geometria situs, 1 0 Irreducible Group relative to carrying a non-bounding r-cycle, of n-chains o f K ove r G, 123 207 of n-cycle s o f K ove r G f 123 relative to carrying an r-cycle non-bounding on Af , 20 7 Hausdorff axiom s Irreducibly connected , 2 1 First, 2 Isomorphic complexes , 89 f Second, 5 Third, 6 Join, 12 7 Fourth (separatio n axiom) , 69 Jordan-Brouwer separatio n theorem , 1 4 ff, First countability axiom , 73 52, 63, 217, 894 Hausdorff space , 69 Converse of , 29 6 ff, 307 ff Homeomorphic, homeomorphism , 8 Jordan curve , 31 Homologous cycles, 12 4 Jordan Curv e Theorem, 1 3 ff, 44, 52, 63, 68, Homology, 11 , 14, 55, 124 88, 211, 214, 217, 286, 290, 353 base, 13 6 Converse of , 67 , 298 relation, 55, 124 relative, 13 1 Klein bottle, 125 , 246 unrestricted, 16 9 Kronecker inde x ( = Ki) , 56 f , 12 2 Homology group, 12 3 ff approximately o n a set, 17 2 Limit point, 2 Cech, 130 , 147 Limit superior (= Li m sup), 102 compact, 24 6 Lindelof theorem , 7 2 examples o f mod 2, mod m , etc., 12 4 ff Linear graph, 1 1 infinite, 24 8 Linear independenc e infinite fundamental , 25 8 relative to homology (= lirh) , 55, 124, 126, invariance of , fo r a complex , 12$ , 146 136 mod L o n M, 132 , 166 relative t o cohomolog y ( = lircoh) , 16 5 of a euclidean complex , 55 Linear isomorphism o f vector spaces , 13 6 of an open subset o f S n, 5 8 Link, 62 , 266 of K ove r G , 123 of a simple chain o f sets, 33 relative, 13 1 irreducibly, 29 6 Homotopic to zero, 287 Linking integral, 1 0 Homotopy Linking, theorems on, 266 ff local connectedness, 19 9 Local co-connectedness, 18 9 ff manifolds, 28 7 characterization of , 19 2 Locally connected spaces, Imbed, 9 characterizations of , 102 , 104 , 10 6 ff, 224, In (a s applied to chains being in an open set), 227, 234 , 318, 320 ff, 339 ff 150 Local connectedness, 12 , 40 Incidence number, 12 1 Local connectedness in higher dimensions, 176 Incident, 11 , 120 characterizations of , 178 , 193 , 197 , 210 , Indexed systems, 14 7 229 ff, 233, 238 Infinite fundamenta l homolog y group, 258 Local connectivity numbers , 190 , 192, 291 ff, Infinite manifold , 296 characterization o f 2-dimensional, 28 0 Local non-r-cut point, 22 8 Interior, 1 7 Local separating point, 276 point, 1 7 Locally arcwise connected, 81 Intersection o f sets, 1 Locally r-avoidable, 218 Inverse systems, 14 7 almost, 23 1 Irreducible continuum , 20 9 sets, 238 400 INDEX

Locally orientable , 28 1 Peano space , 13 , 76 Locally peripherall y countabl y compact , 2 9 Perfectly normal , 16 8 Locus o f concentration , 20 5 Perfectly separable , 7 0 Phragmen-Brouwer property , 4 7 Manifolds. Th e variou s type s ar e liste d ac - of S n, 60 , 68 , 24 2 cording t o descriptiv e terms , suc h a s Poincare space , 24 5 "generalized," "regular, " etc. ; se e als o Point set , 2 , 12 , 1 5 15, 1 6 Positional invariant , 13 , 290 2-Manifold, 9 4 ff Product characterizations of , 95 , 22 1 ff , 225 , 272 , of chains , 15 3 280, 286 , 349, 374 ff. of cycl e and cocycl e o f a space , 15 7 closed, 9 5 space, 3 7 infinite, 9 5 Projection, 12 9 Mappings, 5 ff Projective plane , 24 6 bicontinuous, 8 Property S , 10 6 closed, 7 0 weak, 33 9

continuous, 7 Property S n , 235 topological, 8 duality of , 31 6 ff Metric space , 4, 1 3 rel. G, 235 Metrizable, 7 1 weak, 34 2 Metrization theore m (Urysohn) , 7 2 q-equi valence, 1 8 Neighborhoods Quasi-closed curve , 3 2 defining syste m of , 3 Quasi-component, 1 9 equivalent system s of , 3 relation o f homolog y to , 14 1 ff equivalent system s o f relativ e t o a poin t Quasi-locally connected , 4 0 set, 7 2 of a point , 2 Real numbers, spac e of , 3 Non-cut point , 1 0 Refinement, 12 9 Nondegenerate set , 2 adjusted t o a set , 17 2 Non-r-cut point , 21 8 closure, 13 3 almost, 23 1 normal, 14 0 local, 22 8 star-, 13 3 Norm o f a chain-realization , 17 8 Regular, 10 5 Normal, 4 9 n-manifold, 28 3 completely, 5 0 Normal sequenc e o f refinements , 21 8 S, see Property S . Nucleus, 12 9 Scalar produc t o f chains , 15 9 Schoenflies extensio n theorem , 9 4 On (a s applied t o chains o n a poin t set) , 13 1 Schoenflies-Moore Theorem , 117 , 32 3 Open Semi-locally-connected, 233 , 33 3 arc, 2 7 Semi-n-connected, 16 7 set, 5 Separable, 2 4 Order o f a covering , 19 5 perfectly, 7 0 Orientable, 125 , 246, 24 9 Separate, To , 1 0 Orientability, 24 9 Separate sets , separated sets , 8 Orientation, 11 , 12 0 multiwise, 2 0 pairwise, 2 0 Pair, 23 4 Sequential limi t poin t ( = sip) , 7 3 Peanian = havin g th e propertie s o f a Pean o Sets, 1 space (q.v. ) Cantor theor y of , 1 1 Peano continuum , 13 , 69, 7 6 finitely additiv e collectio n of , 7 0 See "Locall y connecte d spaces " Set-theoretic method , 1 2 INDEX 401

Simple chai n o f sets , 3 3 Topology, 8 Simple chai n theorem , 3 3 algebraic, 12 , 5 2 Simple close d curve , 3 1 combinatorial, 1 1 77-alteration of , 11 5 Torhorst theorem , 114 , 119 , 325, 333, 340 Simplex, 12 0 Torsion, 12 5 Homology group s of , 127 coefficient of , 12 5 Simplicial mapping, 12 7 Totally disconnected , 1 3 Homomorphisms induce d by , 12 8 Triangulation theorem , 9 8 Simply n-connected , 16 8 Two-sidedness, 12 6 Smooth, 36 9 Space, 2 Unicoherence, 4 7 Irreducibly connecte d abou t a subset , 2 1 of the n-sphere , 6 0 Space-filling curv e problem , 1 2 Uniform loca l co-connectednes s ( = r-coulc) , Span, 8 8 190, 29 2 ff 1-Sphere Uniform loca l connectednes s Characterization of , 3 1 ff, 43 ff, 67, 11 4 ff, in dimensio n r (= r-ulc) , 65, 29 2 221 ff, 225, 271, 298, 307, 365, 376 in the sens e o f Cech, 17 8 2-Sphere, 8 7 ff in set-theoretic sens e ( = ulc) , 65 , 10 9 Characterization of , 88 , 22 0 ff, 280 , 307 , of domains complementary t o a fc-sphere i n 365, 374 , 376 ff the n-sphere , 6 6 31, n-Sphere, 3 8 of domain s complementar y t o submani - open subset s of , 5 8 folds o f a manifold , X subdivision of , 5 2 of neighborhood s i n a Pean o space , 7 7 Spherelike, 24 4 of ope n subset s o f a manifold , X Spherical neighborhood , 4 weak ( = r-wulc) , 34 4 Star-finite, 12 2 Union o f sets , 1 Star, 12 0 Universal coefficien t group , 12 6 Subdivision derived, 5 2 elementary, 5 2 Vector space , 13 5 of a chain , 5 8 Vertex of a euclidea n complex , 5 7 of a covering , 12 9 Subspace, 3 of a simplex , 12 0 Successor, 13 8 Weak Hausdorf f space , 4 1 Topological invariant , 8 Weak separatio n axiom , 1 7 Topological property , 8 Weakly locall y connected , 4 0 AUTHORS CITE D

Alexander, 12 , 14 , 52, 67, 95, 125 , 199 , 270 , Lefschetz, 12 , 15, 39, 126 , 175 , 199, 244, 269, 312, 315, 383 270, 283, 289 Alexandroff, 14 , 34 f, 36 f, 39, 68, 98, 125, 207, Lennes, 39 209, 243, 269,270, 272,315,353 ff, 371, 377, Listing, 1 0 383 Mazurkiewicz, 12 , 68 ff, 98, 99 Ay res, 11 9 Meray, 10 , 188 Begle, 199 , 251, 269, 270, 315, 383 Menger, 286, 382 Bernstein, F., 9 Moore, R . L. , 1 3 f , 3 6 f , 39 , 67, 98, 99, 119 , Betti, 1 1 f 307, 315, 316, 323, 336 ff, 345 Bing, 98 Mullikin, 68 Brouwer, 1 3 ff, 68, 244, 294, 315 Nobeling, 382 Butcher, 19 9 Peano, 12 , 69, 98, 99 Cantor, G. , 10 , 11, 188, 337 Poincare, 11 , 12, 125, 245, 269, 383 Cauchy, 10 , 188 Pontrjagin, 269 , 270, 315 Cech, 15 , 129 ff, 146, 175, 199, 224, 243, 269, Riemann, 11 , 12, 1 5 f 270, 289, 315, 383 Schoenflies, 1 2 ff, 39, 67 ff, 116, 245, 316, 323, Chittenden, 1 3 353, 377 Cohn-Vossen, 125 , 246 Seifert, 24 5 Eilenberg, 17 5 Sierpinski, 9 f , 119 , 238 Euler, 1 1 Steenrod, 12 6 Frankl, 14 , 270 Swingle, 68 Frechet, 8 f , 3 6 f Tait, 1 1 Gauss, 1 0 Gawehn, 286 Threlfall, 24 5 Gehman, 26 f , 2 7 f , 3 9 Tietze, 12 , 125 Hahn, H., 12 , 67, 69, 98, 99 Torhorst, 119 , 325 Hausdorff, 2 , 10, 16, 39, 67, 100, 188 Tucker, 125 , 246 Hilbert, 125 , 246 Urysohn, 18, 34 f, 3 6 f, 98, 229 ff, 234, 243 Hurewicz, 382 Vaughan, 199 , 280, 289 Jordan, C, 12 , 69 Veblen, 12 , 125, 246, 269 van Kampen, 67, 98, 243, 244, 245, 282, 283, Vietoris, 14 , 188, 199 286 Wallace, 17 5 Kaplan, S., 175 , 199, 243, 270 Wallman, 382 Kerekjart6, 94 , 125 ff, 246 Weber, 1 1 f Kirchhoff, 1 1 Whitney, 175 , 243 Klein, F., 8 f, 12 5 Whyburn, G. T., 67,98, 99,116,119,233, 277, Kline, 33 f , 119 , 243 316, 333 ff, 353 ff Knaster, 2 0 f , 2 7 f, 39 , 11 9 Wilder, R. L., 20 f, 39, 67 ff, 98,119,199, 243, Kronecker, 12 2 269, 289, 290, 315, 352, 377, 380 Kuratowski, 9 f, 2 0 f , 2 7 f, 39 , 67, 119, 316 Young, G. , 9 8 Lebesgue, 19 6 Zippin, 88, 92

402 ERRATA In eac h instance, the firs t numbe r refer s to the page. "Lin e — n" refer s to th e nth lin e from th e botto m o f the page . 62 Lin e 23. Insert "o f | Zn~2 | " afte r "(n - r)-cells " 76 Lin e -21. Chang e "Lemm a 2.4 " to "Lemma s 2. 3 and 2.4 " 128 Lin e 17 . Inser t "a s i n 6.3" after "K'" n 158 Lin e 20. Inser t "an d C-cycl e z " afte r "H Q(S)" 180 Lin e 15 . Inser t "(23* n)" after secon d "z" a s well as befor e "P " 191 Line s 5 and 7 . Afte r "all " insert "arbitraril y small " 206 Lin e 1 . Befor e "then" inser t "such that jor some closed set K containing M, yr ~ 0 mod if," ; an d befor e "such" inser t "and contained in K" 237 Lin e —10 . Befor e "base" insert "interio r (relativ e t o th e .r*/-plane ) o f the " 283 Lin e 23. Inser t "smal l enough " befor e "neighborhood " 292 Lin e -13. Th e exponen t o f "g" shoul d b e "n - r - 1 " 303 Lin e 8 . Inser t "ulc k" befor e "open" 304 6. 1 Lemma. Inser t "compact" befor e "space" 327 3. 1 Definition ; 3. 2 Definition . Inser t "an d Q is compact " befor e comma . 368 Lin e 1 . Befor e the second period inser t "an d M 0 = {(0 , 0, z) | 0 < z S 11 " 389 Betwee n "Moore , R . L. " and "Poincare , H." inser t "MTJLLIKIN, A . [a] Certain theorems relating to plane connected point sets, Transactions o f the America n Mathematica l Society , vol . 2 4 (1922) , pp . 144-162 "

403