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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
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F Academie de s Sciences , Paris , vol . 17 8 (1924) , pp . 185-187 . 3. Darstellung der Grundziige der Urysohnschen Dimensionstheorie, Mathematisch e
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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 .
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Ayres, W . L.. 1. On continua which are disconnected by the omission of any point and some related
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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 ,
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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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Hamilton, O. H. 1. Fixed points under transformations of continua which are not connected im kleinen t
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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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