Analytic Topolog y

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American Mathematica l Societ y

COLLOQUIUM PUBLICATIONS

Volume 2 8

Analytic Topology Gordon Thomas Whybur n

American Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/coll/028

International Standar d Seria l Numbe r 0065-925 8 International Standar d Boo k Number 0-8218-1028- 6 Library o f Congres s Catalog Card Numbe r 63-2179 4

Copyright © 194 2 by the American Mathematica l Societ y Reprinted wit h corrections 197 1

Printed i n the United State s of America All rights reserved except those granted t o the Unite d State s Governmen t

This book may not be reproduced i n any form withou t the permission o f the publishe r The paper used in this book i s acid-free an d fall s within th e guidelines

established t o ensure permanence and durability . @ 11 10 9 8 76 9 7 96 95 94 93 92

PREFACE

The materia l her e presented represent s an elaboration o n my Colloquium Lec -tures delivere d before th e America n Mathematica l Societ y a t it s September , 1940 meetin g a t Dartmout h College . Th e result s o f som e o f m y ow n effort s together w rith a selection o f those of other mathematicians relativ e to the subjec t chosen ar e presente d i n wha t i s intende d t o b e a coheren t an d approximatel y self-contained exposition , frame d i n th e familia r topolog y o f separabl e metri c spaces. N o attemp t i s mad e a t comprehensiv e coverag e eithe r o f th e know n work embraced b y th e titl e o f the boo k o r of th e wor k of others , or even myself , which ma y b e closel y relate d t o tha t included .

I wis h t o expres s m y appreciatio n t o th e America n Mathematica l Societ y fo r the opportunit y o f deliverin g th e lecture s an d publishin g i n it s Colloquiu m Series. Thank s ar e du e als o t o th e Waverl y Pres s fo r it s carefu l an d sympa -thetic handlin g o f th e manuscript .

On th e persona l side , i t ha s bee n m y privileg e an d goo d fortun e t o stan d i n the middl e groun d betwee n distinguishe d teacher s o n th e on e han d an d a grou p of distinguishe d student s an d associate s o n th e othe r an d receiv e stimulu s an d inspiration fro m both . O f th e former , th e influenc e o f R . L . Moor e wil l b e apparent an d hi s invaluable contributio n i n thi s wa y i s gratefully thoug h inade -quately acknowledged . O f th e othe r group , to o numerou s t o mention , Huber t A. Arnold , M . Garci a an d Pau l A . Whit e hav e helpe d directl y b y reading an d correcting part s o f the manuscript an d proof. Finally , I wish to acknowledge th e generous assistanc e rendere d b y m y wife, Lucille Whyburn, who contributed ma -terially t o th e conten t an d organizatio n o f th e lecture s an d manuscrip t an d assisted greatl y i n th e preparatio n o f bot h an d whos e unfailin g encouragemen t transcends al l attempts a t evaluation .

G. T . WHYBUR N

CHARLOTTESVILLE, VA .

November, 194 1

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TABLE O F CONTENTS CHAPTER PAG E

I . INTRODUCTOR Y TOPOLOG Y

1. Operation s wit h set s 1 2. Topologica l space s 1 3. Ope n an d close d sets . Compac t sets . Separabilit y 3 4. Coverin g theorem s 4 5. Metri c spaces . Metrizatio n Theore m 5 6. Diameter s an d distance s 9 7. Superio r an d inferio r l imits . Convergenc e 1 0 8. Connecte d sets . Well-chaine d set s 1 3 9. Limi t Theorem . Application s 1 4

10. Continu a 1 5 11. Irreducibl e continua . Reductio n Theore m 1 7 12. Continu a o f convergence . Locall y connecte d set s 1 8 13. Semi-locally-connecte d sets . Regula r set s 1 9 14. Locall y connecte d set s 2 0 15. Propert y S. Uniforml y locall y connecte d set s 2 0

I I . CONTINUOU S TRANSFORMATIONS . JUNCTIO N PROPERTIE S O F LOCALL Y CONNECTE D

SETS

1. Continuou s transformation s 2 4 2. Complet e spaces . Extensio n o f transformation s 2 7 3. Junctio n propertie s o f connecte d an d locall y connecte d set s 3 0 4. Mappin g theorem s 3 3 5. Arcwis e Connectednes s Theore m 3 6

I I I . C U T POINTS . NON-SEPARATE D CUTTING S

1. Fundamenta l preliminarie s 4 1 2. Non-separate d cutt ing s 4 4 3. Cu t points . Orde r theorem s 4 9 4. Propertie s o f th e se t E(a, b) - f a + b 5 0 5. Bore l se t classificatio n o f th e cu t point s 5 2 6. Non-cu t points . Propertie s o f simpl e arc s 5 4 7. Simpl e close d curve s 5 7 8. Separatin g point s 58 9. Loca l separatin g point s 6 1

IV. CYCLI C ELEMEN T THEOR Y

1. Conjugat e points . Simpl e links . £Vset s 6 4 2. Cycli c element s 6 6 3. A-sets 6 7 4. Continu a o f convergenc e 7 0 5. Cycli c chain s 7 1 6. / / -set s 7 2 7. Cycli c chai n development . Imbeddin g Theore m 7 3 8. Noda l sets . Node s 7 7 9. Cycli c connectednes s 7 7

10. Som e equivalence s 8 0 11. Applications . Cyclicl y extensibl e an d cyclicl y reducibl e propertie s 8 1 12. Degre e o f multicoherenc e 8 3

Vll

viii TABL E O F CONTENT S

CHAPTER PAG E

V. S P E C I A L T Y P E S O F C O N T I N U A

1. Dendrite s 8 8 2. Hereditaril y locall y connecte d continu a 8 9 3. Rationalit y o f the hereditarily locall y connecte d continu a 9 3 4. Regular , rationa l an d n-dimensional continu a 9 6 5. Classificatio n o f curves 9 8

VI. PLAN E CONTINU A

1. Jorda n Curv e Theore m 10 0 2. Phragmen-Brouwe r Theorem . Torhors t Theore m 10 5 3. Plan e Separatio n Theorem . Application s 10 8 4. Accessibility . Region s an d their boundarie s I l l 5. Characterizatio n o f the sphere 11 4

VII. SEMI-CONTINUOU S DECOMPOSITION S AN D CONTINUOU S TRANSFORMATION S

1. Uppe r semi-continuou s collection s 12 2 2. Uppe r semi-continuou s decomposition s 12 3 3. Relation s betwee n uppe r semi-continuou s decomposition s an d continuous

transformations 12 5 4. Particula r kind s o f transformation s 12 7

(4.1) Monoton e 12 7 (4.2) Non-alternatin g 12 7 (4.3) Interio r 12 9 (4.4) Ligh t 13 0 (4.5) A-se t reversin g 13 1

5. Semi-close d set s an d collections 13 1 6. Nul l collection s 13 4 7. Homeomorphis m o f the original an d the image se t 13 5

VIII. GENERA L PROPERTIES . FACTORIZATIO N

1. Preliminarie s 13 7 2. Characterization s o f monotone an d non-alternating transformation s 13 7 3. Composit e transformations . Produc t theorem s 14 0 4. Factorizatio n 14 1 5. Retraction s 14 3 6. Th e mappin g o f cycli c element s an d A -sets unde r monoton e an d non -

alternating transformation s 14 4 7. Interio r transformation s 14 6 8. Quasi-monoton e transformation s 15 1 9. Th e relative distanc e transformatio n 15 4

a. Propertie s o f the transformation 15 5 b. Application s t o plane region s 15 8

10. Irreducibilit y o f transformation s 16 2 IX. APPLICATION S O F MONOTON E AN D NON-ALTERNATING TRANSFORMATION S

1. N on-alternating transformation s o n boundary curve s 16 5 2. Monoton e transformation s o n spheres, cactoids , plane s an d 2-cells 17 0 3. Existenc e theorem s 17 4 4. Extensio n theorem s 17 9

X. INTERIO R TRANSFORMATION S

1. Actio n o n linear graph s 18 2 2. Inversio n o f simple arc s an d dendrites 18 6 3. Inversio n o f Icca l connectednes s an d th e finite-to-one propert y o n 2-

manif olds 18 9 4. Invarianc e o f the 2-manifold propert y 19 1 5. Analysi s i n the small o n 2-manifolds 19 8

TABLE O F CONTENT S b e

CHAPTER PAG E

6. Degree . Loca l homeomorphism s 19 9 7. Analysi s i n the large o n 2-manifolds 20 0 8. Extensio n t o pseudo-manifold s 20 5

XI. EXISTENC E THEOREMS . MAPPING S ONT O TH E CIRCLE

a. Existenc e theorem s 1. Separatio n an d subdivision 20 9 2. Retraction s int o arc s 21 2 3. Retraction s int o simpl e close d curve s 21 3 4. Interio r non-alternatin g mapping s ont o th e interval an d circle 21 8

b. Mapping s ont o th e circl e 5. Equivalenc e t o 1 22 0 6. Homotopy , exponentia l equivalence , essentialit y 22 $ 7. Propert y (b ) and unicoherence 226 8. Th e group s S*, P(X), B(X), an d so on 22 9 9. p(X) an d r(X) fo r locally connecte d continu a 23 5

XII . PERIODICITY . F I X E D POINT S

1. Preliminaries . Invarian t set s 23 9 2. Cycli c elemen t invarianc e 24 1 3. Th e fixed poin t propert y 24 2 4. Almos t periodicit y 24 6 5. Regula r almos t periodicit y 25 0 6. Orbi t decomposition s 25 3 7. Applications . Th e manifold case s 26 2

BIBLIOGRAPHY 26 6

INDEX 27 5

ERRATA 28 1

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INTRODUCTION

As used here the term "Analyti c Topology " i s meant t o cove r those phases of topology whic h ar e bein g develope d advantageousl y b y method s i n whic h con -tinuous transformation s pla y th e essentia l role . I n th e proces s o f evolving , coining o f ag e an d assumin g mor e stabl e form , topology , throug h interactio n with other branches o f mathematics , not only is leaving its mark o n them bu t i s itself adoptin g mor e an d mor e th e languag e an d symbolis m o f th e olde r fields . Thus, fo r example , w e hav e no t onl y a topologica l functio n theor y givin g th e results o f analysi s whic h ar e essentiall y topologica l i n character , bu t als o a function-theoretic topolog y dealin g with topologica l situation s wit h th e ai d an d principal us e o f som e o f th e basi c tool s o f analysis . Withou t drawin g th e line s too sharply o r givin g to o clea r cu t a definition , le t u s sa y i n a genera l wa y tha t analytic topolog y deal s wit h topologica l situation s wit h th e ai d o f analytica l language an d tools , an d t o som e exten t conversely , jus t a s analyti c geometr y handles geometri c situation s b y analyti c methods . I hop e thi s concep t wil l b e made cleare r a s th e treatmen t progresse s an d actua l example s ar e give n illus -trating th e typ e o f relationshi p whic h ha s bee n s o vaguel y defined .

The majo r question s t o b e deal t wit h are , first , th e existenc e o f transforma -tions of various sorts from a space A t o the same or another space B and , second , the analysi s o f th e actio n o f thes e transformation s o n A t o produc e B. Sinc e thus w re ar e dealin g wit h th e transitio n fro m A t o itsel f o r t o somethin g els e possibly quit e differen t topologically , ou r subjec t exhibit s kinshi p wit h earlie r work o n dynamics i n th e Colloquiu m Series . Thi s i s especially tru e o f th e fina l chapter o n periodicity whic h connects directly wit h man y o f th e concepts o f thi s subject a s discusse d b y G . D . Birkhoff . However , th e even close r kinshi p wit h other purely topological treatises , notably that o f R. L . Moore in the Colloquiu m Series and tha t o f K . Menge r on "Kurventheorie" , wil l be too obvious t o requir e comment.

The boo k divide s roughl y int o tw o parts , correspondin g t o th e firs t si x an d last six chapters, respectively . I n the first par t ther e i s developed th e necessar y topological machiner y an d framewor k fo r th e latte r part , whic h i s devote d t o pure analytic topology . Eve n in the second chapter , however , notably in §§3 , 4, there emerg e som e o f th e fruit s o f th e applicatio n o f analyti c o r transforma -tion method s t o topologica l situations . Fo r her e a variet y o f results , som e classic and other s quit e recent , ar e brough t togethe r i n wha t seem s thei r prope r relationship an d derive d i n a simpl e an d nove l wa y fro m on e centra l mappin g theorem.

The book i s meant t o be largely self-contained, a t leas t i n so far a s topologica l developments ar e concerned . I n th e late r stage s som e us e i s mad e o f a fe w notions o f combinatoria l topolog y an d o f th e theor y o f group s withou t an y attempt a t adequat e introduction . Sinc e these appear largel y in end-results an d

xi

Xll INTRODUCTION

applications, ther e seem s littl e nee d o r justification fo r takin g th e spac e t o de -velop the m here .

At the beginning we assume once for all a set o f axioms sufficient t o make al l spaces considere d separabl e an d metrizable . Onc e th e metri c i s introduced , however, attention is no longer focused on the axioms, but rather on the (equiva -lent) standpoint tha t we are operating always in a given separable metric space.

Cross references are given in brackets, with the roman numeral for the chapter followed by the section and number of the theorem, lemma, or corollary referred to, e.g. , [IV , (3.2) ] refers to result (3.2 ) o f § 3 in Chapter IV, which would be the second main resul t i n thi s section . I f onl y th e numbe r i n parenthesi s i s given , as (3.2 ) fo r example, the reference i s to the resul t o f tha t numbe r in the present chapter, i.e. , th e on e bein g rea d a t th e time . T o assis t i n locatin g result s re -ferred to, the chapter number and section number appear at the heading of each double page .

References t o th e literatur e i n th e mai n ar e hel d t o a minimum . Fo r con -venience thes e ar e mad e a t th e end s o f th e chapter s i n th e for m o f author' s name followe d b y numeral s i n brackets referring to his books or papers by tha t number i n th e bibliograph y a t th e clos e o f th e book . I n som e case s on e o r more authors ' name s hav e bee n use d i n connectio n wit h a theore m thoug h b y no means in al l wher e thi s might well , o r possibly should, be done. A consider -able amount o f th e materia l i n the firs t par t i s of such a classical natur e an d so well know n tha t specifi c citation s t o source s i n th e literatur e ar e no t made . Later o n mor e attemp t i s mad e t o cit e th e origina l autho r an d source . I n some cases , also , closel y relate d materia l no t actuall y covere d i s mentione d i n the references .

BIBLIOGRAPHY Adkisson, V. W.

1. (Wit h MacLane, S.) Extending maps of plane Peano continua, Duk e Mathematica l Journal, vol . 6 (1940) , pp. 216-228.

Aitchlson, B . 1. Concerning regular accessibility, Fundament a Mathematicae , vol . 2 0 (1933) , pp .

119-228. Alexander, J . W.

1. A proof of Jordan's theorem about a simple closed curve, Annals o f Mathematics , (2), vol . 2 1 (1919) , pp . 180-181 . A proof and extension of the Jordan-Brouwer separation theorem, Transaction s o f th e America n Mathematica l Society , vol . 2 3 (1922), pp. 333-349.

Alexandrofif, P . 1. Ober die Metrization der im kleinen kompakten topologischen Rdume, Mathematische

Annalen, vol . 92 (1924) , pp. 294-301 . 2. Sur les ensembles de la premiere classe et les ensembles abstraits, Compte s Rendus de

F Academie de s Sciences , Paris , vol . 17 8 (1924) , pp . 185-187 . 3. Darstellung der Grundziige der Urysohnschen Dimensionstheorie, Mathematisch e

Annalen, vol . 9 8 (1928) , pp . 31-63 . 4. Une generalisation nouvelle du theoreme de Phragm&n-Brouwer, Compte s Rendus de

FAcademie de s Sciences , Paris , vol . 18 4 (1927) , pp. 575-577. 6. (Wit h Hopf , H. ) Topologie, I , Berlin , 1935 , Springer . 7. (Wit h Urysohn, P. ) Mtmoire sur les espaces topologiques compacts, Verhande -

lingen de r Akademi e va n Wetenschappen , Amsterdam , vol . 1 4 (1929) , pp . 1-96 . Arnold, H. A .

1. Homology in set-intersections, with an application to r-regular convergence, and on r-regular convergence of sets, Bulleti n o f th e America n Mathematica l Society , abstracts 47-7-32 5 and 47-7-326 .

Aronszajn, N . 1. Ein Urbildproblem, Fundament a Mathematicae , vol . 1 7 (1931) , pp . 92-121 .

Ayres, W . L.. 1. On continua which are disconnected by the omission of any point and some related

problems, Monatshefte fu r Mathemati k un d Physik , vol . 3 6 (1929) , pp . 135-148 . 2. Continuous curves homeomorphic with the boundary of a plane domain, Fundament a

Mathematicae, vol . 1 4 (1929) , pp . 92-95. 3. Concerning continuous curves in metric space, America n Journa l o f Mathematics ,

vol.51 (1929) , pp. 577-594. 4. Continuous curves which are cyclically connected, Bulleti n Internationa l d e

F Academie Polonais e de s Science s e t de s Lettres , Class e de s Science s Mathe -matiques e t Naturelles , A (1928) , pp . 127-142 .

5. Concerning the arc-curves and basic sets of a continuous curve, Transactions o f th e American Mathematica l Society , vol . 3 0 (1928) , pp . 567-578 .

6. Concerning the arc-curves and basic sets of a continuous curve. Second pappr, ibid. , vol. 31 (1929), pp. 595-612.

7. Some generalizations of the Sherrer fixed-point theorem, Fundamenta Mathematicae , vol. 1 6 (1930), pp. 332-336.

8. On transformations having periodic properties, ibid. , vol . 3 3 (1939) , pp . 95-10 3 (reprint).

266

BIBLIOGRAPHY 267

Birkhoff, G . D . 1. Dynamical Systems, America n Mathematica l Societ y Colloquiu m Publications ,

vol. 9 , 1927. Borsuk, K .

1. Einige S&tze uber stetige Streckenbilder, Fundament a Mathematicae , vol . 1 8 (1932), pp. 198-214 .

2. Sur les rttractes, ibid. , vol . 17 (1931), pp. 152-170 . 3. Quelques thforemes sur les ensembles unicoherent, ibid., vol . 1 7 (1931) , pp . 171-209 .

Brouwer, L . E. J . 1. Over de structur der perfecte punktnerzamelingen, Akademie Versammlungen, Amster-

dam, vol. 1 8 (1910) , pp . 833-842 an d vol . 1 9 (1911) , pp . 1416-1426 . Se e als o Pro -ceedings, Akademi e va n Wetensohappen , Amsterdam , vol . 1 4 (1911) , p . 138 .

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3. Vber Abbildungen von Mannigfaltigkeiten, ibid. , vol . 7 1 (1912) , pp . 97-115. 4. Aufzdhlung der periodischen Transformationen des Torus, Proceedings , Akademi e

van Wetenschappen , Amsterdam , vol . 2 1 (1919) , pp . 1352-1356 . Cairns, S . S .

1. The cellular division and approximation of regular spreads, Proceeding s o f th e National Academ y o f Sciences , vol . 1 6 (1930) , pp. 488-490 .

CarathSodory, C . 1. Vber der Begrenzung einfach zusammenhdngender Gebiete, Mathematisch e Annalen ,

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1. Topological immersion of peanian continua in a spherical surface, Annals of Mathe-matics, (2) , vol . 3 5 (1934) , pp . 809-835 .

Cleveland, C . M. 1. Concerning points of a continuous curve that are not accessible from each other, Pro-

ceedings o f th e Nationa l Academ y o f Sciences , vol . 1 3 (1927) , pp . 175-276 . Eilenberg, S .

1. Sur la multicoh&rence des surfaces closes, Odbitk a z e Sprawozdani a z posiedze n Towarzystwa Naukaweg o Warszawskiego , vol . 3 0 (1937) , pp. 109-111 .

2. Transformations continues en circonference et la topologie du plan, Fundament a Mathematicae, vol . 2 6 (1936) , pp . 61-112 .

3. Sur les espaces multicohtrents I , ibid. , vol . 2 7 (1936) , pp. 151-190 . 4. Sur les transformations d'espaces metriques en circonference, ibid., vol . 2 4 (1935) ,

pp. 160-176 . 5. Sur les transformations continues d'espaces mUriques compacts, ibid., vol . 22 (1934),

pp. 292-296 . 6. Sur quelques proprittis des transformations localement homtomorphes, ibid., vol . 2 4

(1935), pp . 33-42 . 7. O n a theorem of P. A. Smith concerning fixed points for periodic transformations,

Duke Mathematica l Journal , vol . 6 (1940) , pp. 428-437. 8. Sur les transformations ptriodiques de la surface de sphere, Fundament a Mathe -

maticae, vol . 2 2 (1934) , pp . 28-41 . Forder, H . G .

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1. Les Espaces Abstracts, Gauthier-Villars , Paris , 1926 . Gehman, H . M .

1. Concerning certain types of non-cut points, with application to continuous curves, Proceedings o f th e Nationa l Academ y o f Sciences , vol . 1 4 (1928) , pp. 431-433.

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268 BIBLIOGRAPHY

Hahn, H . 1. Vber die Komponenten ojfenen Mengen, Fundament a Mathematicae , vol . 2 (1921) ,

pp. 189-192 . 2. Mengentheoretische Characterisierung der stetige Kurven, Sitzungsberichte , Aka -

demie de r Wissenschaften , Vienna , vol . 12 3 (1914) , p. 2433 . Hall, D . W.

1. On a decomposition of true cyclic elements, Transactions o f th e America n Mathe -matical Society , vol . 47 (1940), pp. 305-321.

2. (Wit h Kelley , J . L. ) Periodic types of transformations, Duk e Mathematica l Journal, vol . 8 (1941) , pp . 625-630 .

3. (Wit h Schweigert , G . E. ) Non n-alternating transformations, ibid. , vol . 3 (1937) , pp. 623-626.

4. (Wit h Schweigert , G . E. ) Properties of invariant sets under pointwise periodic homeomorphisms, ibid. , vol . 4 (1938) , pp . 719-724 .

Hamilton, O. H. 1. Fixed points under transformations of continua which are not connected im kleinen t

Transactions o f th e America n Mathematica l Society , vol . 4 4 (1938) , pp . 18-24 . Harrold, O . G.

1. A note on strongly irreducible maps of an interval, Duk e Mathematica l Journal , vol. 6 (1940) , pp. 750-752.

Harry, C . H. 1. Concerning the geometry of acyclic sets, America n Journa l o f Mathematics , vol . 56

(1934), pp. 233-253. Hausdorff, F .

1. Mengenlehre, d e Gruyter , Leipzig , 1927 . 2. Die Mengen G* in vollstdndigen Rdumen, Fundament a Mathematicae , vol . 6 (1924), pp. 147-148 .

Hildebrandt, T . H . 1. The Borel theorem and its generalizations, Bulleti n o f th e America n Mathematica l

Society, vol . 32 (1926), pp. 423-474. Hopf, H .

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Janiszewski, S . 1. Sur les continus irrlductibles entre deux points, Journa l d e Tficol e Pol y technique,

(2), vol . 1 6 (1912) , pp. 7&-170 . Jones, F . B.

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Jordan, C. 1. Cour a"Analyse, Paris , 1893 , 2d edition , p . 92.

Kaluzsay, A. 1. Se e Kerekjarto [4] , p. 103.

Kainpen, £ . R . v . 1. On some characterizations of 2-dimensional manifolds, Duk e Mathematica l Journal ,

vol. 1(1935), pp. 74-93. Kelley, J . L.

1. A decomposition of compact continua and related theorems on fixed sets under con-tinuous transformations, Proceeding s o f th e Nationa l Academ y o f Sciences , vol . 26 (1940), pp. 192-194 .

2. Se e Hal l [2] .

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Ker£kjart6, B . v. 1. Involutions et surfaces continues, Szeged , Act a Litteraru m a c Scientiarum , vol . 3

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Mathematische Annalen , vol . 8 0 (1919) , pp . 36-38 . 4. Topologie, Berlin , 1923 .

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INDEX

Absolute neighborhoo d retract , 21 6 Accessibility, 11 1

from al l sides , 111 regular, 11 1

Acyclic curve, 88 Additive system , 9 7 Arc, 3 6 Arc-curve, 8 0 Arc wise connectedness , 3 6 A -set, 6 7 ff.

invariance of , 14 5 true A -set, 68

Bibliography, 26 6 Borel Theorem , 5 Borel set , 52

classification o f cut points , 52 Boundary, 1 6

curve, 76 , 165 elements o f a plan e region , 160 of a n ope n set , 1 6

Brouwer Fixed Point Theorem, 243 Brouwer Reduction Theorem, 17

Cactoid, 7 6 invariance under monotone mappings, 171

Cantor discontinuum , 35 Cantorian manifold (local) , 149 Cell, 11 6

2-cell, characterization of , 11 9 Characteristic, Euler , 201 Characterization o f the sphere, 114 Closure o f a set , 2 Collection, 1

countable, 1 null, 67 , 134 semi-closed, 13 1 upper semi-continuous , 122

Compact set , 3 Complete

enclosure, 27 locally, 15 8 space, 27

Component, 1 3 orbit, 25 9

Conditionally compac t set , 5 Conjugate points , 64

Connected set , 13 Continua, 1 5

i£-continua, 11 2 generalized, 1 6 hereditarily locall y connected , 8 9 ff. irreducible, 1 7 n-dimensional, 9 6 ff. of convergence , 18 , 70, 81

Convergence of sequence s o f points , 3 of sequenc e o f sets , 10 0-regular, 17 4

Covering theorems , 4 Curve

acyclic, 8 8 arc-, 8 0 boundary, 76 , 16 5 classification, 98 rational, 82-83 , 96 regular, 82-83 , 9 6 simple closed , 57

Cut point , cutting , 41 Cut point-order theorem, 49 ff. Cyclic

additivity o f 6(X) , 234 additivity o f r(X) , 85 set, 10 7

Cyclic chain , 7 1 approximation theorem , 7 3 development, 73 ff.

Cyclic connectedness, 77 ff. theorem, 7 9

Cyclic element , 66 invariance, 24 1 ff.

Cyclicly extensibl e an d cyclicl y reducibl e properties, 8 1 ff.

Decomposition space , 123 Decomposition, uppe r semi-continuous , 12 3 Degree

of a transformation , 19 9 of multicoherence , 8 3

Dendrite, 88 Diameter, 9 Dimensionality, 82-83 , 9 6 Disjoint sets , 1 Distance, 9

275

276 INDEX

Divisibility Theorem, 49

€-chain, 1 3 ^-continuum, 11 2 End point , 6 4 Equicontinuous powers of a transformation,

252 relation to regular almost periodicity, 252

Equivalence o f tw o transformations , 12 7 Equivalence t o 1 , 220 ff.

cyclic extensibility of , 222 on cartesian product spaces, 223

«-set, 31 #o-set, 6 4 Essential transformation , 225 Euler characteristic , 20 1

effect o f interio r transformatio n on , 202, 208

Exponential equivalence , 22 5 Extension of transformations , 28 , 179

interior, 213, 215 monotone, ofnon-alternating , 18 0

Factorization, 134 , 14 1 monotone-light, 14 3

Factor theorem , 14 1 Fixed point , 24 0

property, 242 theorem (Brouwer) , 243

Frontier, F(G), 16

Generalized continuum , 16 Graph (linear) , 18 2

interior transformations on , 182 Groups Sx, PCX), B(X), 22 9 ff. G«-set, 2 8

Hemi-cactoid, 172 invariance unde r monoton e transforma -

tions, 17 3 Hereditarily locall y connecte d continua ,

89 ff. invariance unde r monotone mapping , 13 8 rationality of, 93 ff.

Hereditary system, 97 Hilbert parallelotope , 6 Hilbert space , 6 Homeomorphism, 24

local, 19 9 of origina l an d image sets , 135

Homotopy, 22 5 ff. #-set, 7 2

Hyperspace (o f uppe r semi-continuou s de -composition), 12 3

Imbedding theorems , 33 , 74 Inducible property , 1 7 Interiority a t a point, 149 Interior transformations , 12 9

action o n linea r graphs , 18 2 analysis i n th e larg e of , 20 0 analysis i n th e smal l of , 19 8 effect o n the numbers b(X) an d p(X), 23 2 finite-to-one property , 18 9 invariance o f boundar y curves , 18 5 invariance o f noda l set , 14 8 invariance o f 2-manifolds , 19 1 ff. inversion o f arc s an d dendrites , 18 6 inversion of components , 148 inversion of cyclic elements, 151 inversion o f quasi-components , 14 7 non-increasing o f order , 14 7 on dendrites , 18 5 on invers e set s 14 7 on pseudo-manifolds , 20 5 on simple arcs , 184 on simpl e close d curves , 18 4

Invariant cyclic elements , 242 , 248 set, 24 0 set (completely) , 253 -point, 24 0

Irreducible continuum , 1 7 Irreducible cutting , 4 3 Irreducible transformations , 16 2 ff. Inverse set, 137

Joining of sets by regions, 93 Jordan Curve Theorem, 10 0 ff.

Limit point, 2 superior an d inferior , 1 0 theorem, 1 4

Lindelof Theorem , 4 Locally connecte d continua , 23 ff.

equality o f r(X) an d p(X) for , 238 Local homeomorphism , 19 9 Local separating point, 61 Locally compact set , 15 Locally connected set, 18, 20

inversion unde r interior transformations , 189

uniformly, 2 2

INDEX 277

Manifold can tori an, 14 9 invariance unde r interio r transforma -

tions, 19 1 ff pointwise periodi c mappings on , 262 pseudo-, 20 5 2-dimensional, 19 3

Mapping interior non-alternating , ont o interva l

and circle , 218 monotone, o f locall y connecte d con -

tinuum into regular curve, 219 of boundar y element s o f a plan e region ,

160 of circl e ont o arbitrar y boundar y curve ,

166 of cycli c elements and A-sets, 144 of interva l ont o arbitrar y locall y con -

nected continuum , 3 3 of plan e region onto circle interior, 16 1 of spher e ont o arbitrar y cactoid , 17 5 of 2-cell onto arbitrary hemi-cactoid, 177 onto th e circle , 219 ff. theorems, 3 3 ff.

Metrization theorem , 7 Monotone transformations , 70 , 127

characterizations of , 13 7 ff. effect o n the numbers b(X) an d p(X), 232 of loca l connecte d continua int o regula r

curves, 219 on spheres , cactoids , planes , an d 2-cells ,

170 ff. Multicoherence, degre e of , 8 3

cyclic additivity of , 8 5 Mutually separate d sets , 1 3

Nodal set , 77 , 148 Node, 77 Non-alternating transformations , 12 7

characterizations of , 13 7 ff. on boundar y curves , 16 5 ff.

Non-cut poin t existenc e theorem , 54 Non-separated collection , 4 2 Non-separated cuttings , 4 4 ff. Null sequence (collectio n or family), 67, 134

Orbit, 253 component, 259 continuity of , 25 7 ff. decomposition, 25 3 ff. invariance unde r limi t taking , 25 9

Order of a subset, 48 potential, 45

Ordered set, 41 naturally, 41

Perfect set, 53 Periodicity, 23 9 ff.

almost, 246 regular almost , 250

Periodic transformation , 23 9 almost, 246 elementwise, 248 pointwise, 239 pointwise almost , 246 reduction of pointwise periodic to periodic

on manifolds, 26 2 ff. reduction o f pointwis e periodi c t o regu -

larly almos t periodic, 262 regular almost, 250

PhragmSn-Brouwer Theorem, 105 Plane separation theorem, 108 Point, 1

condensation, 4 fixed, 240 limit, 2

Points o f interiorit y o f a transformation , 149

Product cartesian or topological, 6 of sets , 1 theorems for transformations, 14 0

Property S, 20 , 111 Pseudo-manifold, 20 5

interior transformation on , 205

Q«, QL, 6 Quasi-component, 9 0 Quasi-monotone transformation, 151

characterizations of , 152-15 3 effect o n degree of multicoherence, 153 invariance under , 153-15 4

Rational curve , 82-83, 96 ff. 138 Reduction Theorem, 17 Region, 20 Regions and their boundaries, 111 ff., 158 ff. Regular

chain, 36 convergence, 174 curve, 82-83, 96 ff., 138 set, 19 space, 2

278 INDEX

Relative distanc e transformation , 15 4 Retract, absolut e neighborhood , 216 ff. Retractions, 70 , 143

non-alternating interior , of locally connected continua into arcs,

212 of locall y connecte d continu a int o

simple close d curves , 21 3

Saturated collection, 45 Semi-closed (set , collection) , 131 ff . Semi-locally-connected set , 1 9 Separability, 3 Separating point , 5 8

order theorem, 60 Separation, 13 Sequence, 1

convergence of , 3,1 0 null, 67

Sets compact, 3 countable, 1 dense, 30 G«-,28 locally compact , 15 locally connected , 18 nondegenerate, 30 open and closed, 3 operations with , 1 perfect, 53 regular, 19 semi-closed, 13 1 semi-locally-connected, 1 9 totally disconnected , 34

Simple arcs , interior transformation s on , 184 pointwise periodi c mapping s on , 264 properties of , 5 4 ff .

Simple close d curves , 57 interior transformatio n on , 18 4 pointwise periodic mappings on, 264 separation of plane or sphere by, 100 ff.

Simple link, 64 Space

complete, 27 Hilbert, 6 metric, 5 normal, 6 of continuou s mappings , 27 perfectly separable , 2 regular, 2 topological, 1 topologically complete , 27

Sphere in variance unde r monoton e transforma -

tion, 17 0 topological, characterization of , 11 4

0-curve (Theta-curve) , 10 4 separation o f plan e by , 10 4

Three poin t theorem , 7 9 Topological

space, 1 sphere, 114 transformation, 2 4

Topologically equivalen t transformations , 127

Transformation, 2 4 Aset reversing , 70 , 131 composite, 140 continuous, 24 interior, 129 inverse o f a , 24 light, 13 0 monotone, 70 , 127 non-alternating, 12 7 one-to-one, 24 particular kind s of , 12 7 ff. periodic, 239 quasi-monotone, 151 relative distance , 15 4 retracting, 70 , 143 topological, 2 4 topological equivalence of one to another,

127 uniformily continuous , 25

Unicoherence, 82 about a simple close d curve , 213 and property (b) , 226 fif. cyclic extensibility and reducibility of , 82 invariance unde r monoton e transforma -

tions, 13 8 of cartesian product of continua , 228

Uniformily locall y connecte d sets , 2 2 plane regions , 161

Upper semi-continuou s collections, 122 decomposition, 12 3 decomposition, relation s t o continuou s

transformations, 12 5 ff.

Ve (X), 12

Well-chained set , 1 3

Young, W . H. , Theore m of , 5 3

NOTES

1. Th e conclusions in (7.3) o n p. 14 7 hold without any restriction on A pro-vided R i s assume d conditionall y compact . Th e thu s broadene d (7.3 ) the n yields the further corollary.

(7.32) Local compactness is invariant under interior transformations. Local compactness of A also should be assumed in (7.31). 2. Th e theorem in (3.1) o n p. 18 9 may be strengthened by allowing A t o be

any generalized continuum instead o f restricting i t to continua. Thi s improved form of (3.1 ) applies more directly in the proof of (3.42) on page 191 and facili-tates th e reading o f tha t proof.

3. I n th e proof o f (ii ) o n pgs. 194-195 , there i s no advantage i n having th e set M connected; and some simplification results if it is taken merely to be com-pact and locally connected .

4. I n the proof of (iv) on p. 195, the points r and s may and should be taken on the arc uqv in the order uf r, q, a, v.

5. Th e proof of (v ) o n p. 195 is sketchy and difficult t o follow. I t should be replaced by the following detaile d argument :

By (iv) , every poin t o f K i s of orde r : § 2 . Suppose , first, that som e x € K is o f orde r 1 i n K. Le t V b e a n €/2-neighborhood o f x i n B suc h tha t F(V) • K = b 6 K. B y (ii) and (iv) there exists an €/2-neighborhood W of b in B whose boundary i s a simple arc rs with rs-K = r + s and so that W does not contain x. Sinc e x is of order 1 in K, i t is clear that for t sufficiently smal l one end point o f rs, say r , i s in V and the othe r in B — Vt a s otherwise we could obtain fro m V and W an €-neighborhood o f x with boundary disjoint fro m K. Let t be an interior poin t o f rs such tha t rt C V . The n i f T denote s th e su m of a finite numbe r o f locall y connecte d subcontinu a o f B s o chose n tha t T 3 F(V) - W-F(V), T-(K + x) = 0 , b(T + V) < 2*/Z and M - T + ts, then M i s a compact locall y connecte d se t 6-separatin g x in B and such tha t MK = s. Bu t then the reasoning given under (ii) and (iv) shows that M con-tains a simple arc X both of whose end points belong to K and this is impossible.

Now in case some a G K wer e of order 0, we need only choose V as above so that F(V) -K = 0 and let M be a compact locally connected set in B containing JF(F), disjoint from K + x and so that 5(V + M) < « . The n by the reasoning under (ii ) an d (iv ) w e would obtai n i n M a simple close d curv e X which , b y (iii), would show that x is a regular point, contrary to x G K .

6. Th e set f~l{K) use d under (viii ) o n p. 19 6 is necessarily a finite graph as there stated because it contains only a finite number of simple closed curves and has no end points.

7. I n the proof o f (4.6 ) o n p. 249, i t i s tacitly assume d tha t th e mapping /n

is pointwise almos t periodi c a t p. Thi s can be verified fairl y easil y under the assumptions made in (4.6) . Also , it i s a consequence o f more general proposi-tions along this line established late r by W . H. Gottschalk . See , for example, Bulletin of the American Mathematical Society, vol. 50 (1944), p. 223.

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8. Inlin e 3 of the proof o f (5.2) , p . 71, to see that A is separated betwee n a and 6 , le t A = A \ + A 2 be an y separatio n wher e 6 £ A 2. I f a £ A' 2, the n since A \ i s nonempt y s o tha t no t al l point s o f E(a, b) ca n b e i n A 2, ther e is a point c of E(a, b) in A^ . Then i f Q is the component o f M — c containin g 6, the set X b = Q • A2 is both ope n an d closed i n A an d doe s no t contai n a.

9. Th e autho r i s indebte d t o Dr . R . J . Bea n fo r pointin g ou t tha t th e result (7.2) , p . 136 , in earlie r edition s o f thi s boo k i s incorrectl y worded . The word s for each b £ B shoul d b e inserte d betwee n " tha t " an d "there " in th e secon d lin e o f th e statemen t o f thi s resul t i n th e earlie r editions .

10. I n the proof o f (3.4) , p . 216, to se e tha t M i s no t unicoheren t abou t J, w e reason a s follows. Fo r each poin t p o f th e intersectio n o f A (resp . B) with th e ope n ar c ayb let B p b e a regio n i n M abou t p o f diamete r < 1/ 3 the distanc e fro m p t o axb + B (resp . A) an d le t Ft'* b e a finite unio n o f the Rp covering ayb • (A + B) - V 1/n(a + b). Then th e sets W = H - X : = 1 / ? ; , K' = K +^n=iR'p, ar e close d an d w e hav e

H' . J = axb, K' -J = ayb, M = H' + A' , W - A' £ V d/3(H - A)

where d = p(A, B). Thu s a an d 6 lie in differen t component s o f H' • K\

ERRATA

p. 19 . Substitut e th e followin g fo r th e final paragrap h o f th e proo f o f (13.2) .

Consider first th e cas e o f compac t M. No w le t d an d D\ b e component s o f Mi = M — Vt/2(x) intersectin g bot h F[V t/2(x)] an d M — V t(x) an d le t

M1 = C\ + Dl

be a separatio n o f Mi betwee n Ci an d D\. The n fo r a t leas t on e o f th e set s Ci + V e/2(x) an d D ? + V t/2(x), sa y Xi, i t mus t b e tru e tha t fo r ever y i > 2, infinitely man y component s o f Xi — Vt/i(x) intersec t M — V t(X). W e suppos e this hold s for Xi = C ? + V t/2(x) an d le t Ki b e a component o f Di — D r F 3</4(x) intersecting M — V€(x).

Then le t C% an d D 2 b e component s o f M 2 = Xi — F«/3(x) intersectin g bot h F[Vt/3(x)] an d M - V t(x) an d le t

M2 = C° 2 + D° 2

be a separatio n o f M 2 betwee n C 2 an d D 2. Proceedin g a s before w e find X2 = C 2 + V t/i(x) [o r — D 2 + F </3(x)] s o tha t infinitel y man y component s o f X2 — Vt/i(x), i > 3 , intersec t M — ^ ( x) an d le t K 2 b e a componen t o f D2 — Z>2-F 2e/3(^) intersectin g M — V t(x). Continuin g thi s process indefinitel y we obtai n a n infinit e sequenc e K\ , K 2, • • • o f disjoin t continu a eac h inter -secting M - V t(x), wit h K n-F[V2e_(x)] ^ 0 an d

n+l

n+l

Thus if [K n{] is a convergent subsequence of [/C„ ] with limi t K, w e have Kc I I C ° so tha t KK n = 0 fo r al l n ; an d thu s X i s a continuu m o f convergenc e o f M containing x.

Next suppose , contrar y t o ou r theorem , tha t x i s o n n o continuu m o f con -vergence o f M. The n M i s locall y connecte d a t x and , accordingly , fo r an y e > 0 , x i s interio r t o th e componen t C e o f V t(x) containin g x . Suppos e € chosen s o that C « is compact . B y th e compac t cas e treate d above , C t mus t b e semi-locally connecte d a t x. Thu s w e ca n find a n arbitraril y smal l < r < e such that V ff(x) a C ( an d s o tha t C € — Va{x) = i V ha s onl y a finite numbe r k o f components. Bu t the n M — Va(x) = N + M — Ct ca n hav e a t mos t k com -ponents since M — Ct and F<r(x ) are separated and M — N = (A f — C« ) + ^ ( x ) . Whence, M i s s.l.c . a t x contrary t o hypothesis .

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