FOUNDATIONS O F POIN T SE T THEOR V This page intentionally left blank http://dx.doi.org/10.1090/coll/013 AMERICAN MATHEMATICA L SOCIETY COLLOQUIUM PUBLICATIONS , VOLUME XIII FOUNDATIONS O F POINT SE T THEOR Y R. L . MOORF . Revised Edition AMERICAN MATHEMATICA L SOCIET Y PROV1DENCE, RHOD E ISLAN D FIRST EDITIO N PUBLISHE D 193 2 COMPLETELY REVISE D AN D ENLARGE D EDITIO N 196 2 2000 Mathematics Subject Classification. Primar y 54-XX . Library o f Congress Catalo g Car d Numbe r 62-832 5 International Standar d Boo k Numbe r 0-8218-1013- 8 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t libranes actin g fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o copy a chapte r fo r us e i n teachin g o r research . Permissio n i s grante d t o quot e brie f passages fro m thi s publicatio n i n reviews , provide d th e customary acknowledgmen t o f the sourc e i s given. 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Visit the AM S home page at URL : http://ww.ams.org / 12 11 10 9 8 7 0 6 05 04 03 02 01 TABLE O F CONTENT S Pag* Preface ........... vi i Introduction .......... i x Chapter I . Consequence s o f Axiom s 0 and 1 ... ] Chapter II . Consequence s o f Axioms 0 , 1 and 2 . .8 4 Chapter III . Consequence s o f Axioms 0 , 1 , 2 , 3 and 4 . .14 1 Chapter IV . Consequence s o f Axioms 0 and 1- 5 .... 16 2 Chapter V . Uppe r semi-continuou s collection s 1. O n the basi s o f Axiom s 0 , 1 ' an d C . 27 3 2. O n the basi s o f Axiom s 0 , ]' , C, 2 . 3 , 4 and 5 .31 1 3. Equicontinuou s collection s o n the basi s o f Axiom s 0 and 1 .33 1 Chapter VI . Consequence s o f Axiom s 0 , 1 , 2 , 3 , 4 , 5i , Ö2 , 6 and 7 . 33 9 Chapter VII . Concernin g topologica l equivalenc e an d th e intro - duction o f distance ...... 3o 3 Appendix . .37 8 Bibliography 38 2 Glossary . .41 7 This page intentionally left blank PREFACE In thi s revise d editio n ther e i s agai n presente d wha t ma y b e roughl y termed a largel y Belf-containe d treatmen t o f the foundation s o f continuity . or point set-theoretic , analysi s situ s (topology) . AU the numbered proposition s o f Chapter 1 are proved o n the basis o f two axioms ( 0 and 1 ) that hol d true i n a very large dass o f Spaces including man y Spaces that ar e not locall y connected . I f thes e axiom s hol d tru e i n a spac e 2 an d S' i s any inne r limitin g se i i n 2 then , unde r a suitabl e interpretatio n of th e wor d region, the y hol d tru e als o i n a spac e i n whic h point mean s point o f S'. The numbered proposition s o f Chapter II hol d true i n all locally connecte d Spaces tha t satisf y th e first thre e axioms . Thi s das s o f space s include s Euclidean space s o f any finite numbe r o f dimensions, Hubert space , an d th e spaces o f infinitel y man y dimension s whic h Freche t designate s b y th e symbols D w an d E w. I f Axiom s 0 . 1 and 2 hold true i n a space 2 an d S' i s a locally .connected inne r limiting set in 2 then . under a suitable interpretatio n of the word region , they hol d tru e als o i n a space i n whic h the wor d point i s restricted t o mea n poin t o f S'. lt i s shown , i n Chapter s II I an d IV , that , o n th e basi s o f Axiom s 0,1,2,3.4 an d 5 , it i s possible to prove a very considerable portion o f the well known topological propositions of the plane. Nevertheles s there exist space s which satisfy thes e axioms. and therefore in which al l the numbered theorem s of thes e chapter s hol d true , bu t whic h ar e neithe r metric , locall y compac t nor separable an d i n which, moreover , i t i s not even tru e that i f P i s a poin t of a domai n D the n ther e exist s a domai n lyin g i n D, containin g P , an d bounded b y a simpl e close d curve . Chapter V is concerned particularly with upper semi-continuous collections . In Chapter VI there is formulated a set of axioms having the property tha t every compac t spac e that satisfie s al l o f them i s topologically equivalen t t o a sphere , whil e every noncompac t on e is topologically equivalen t t o a plane. In hi s paper Concerning B. L. Moore's Axiom 5 , F. Burto n Jone s showe d that i f Axiom 5 ' denotes the axio m obtaine d b y replacin g simple closed curvc by compact continuum i n the Statement o f Axiom 5 of the first editio n o f this book the n th e origina l Axio m 5 i s a consequenc e o f Axiom s 0- 4 an d 5' . This Axiom 5 ' i s the Axio m 5 of the present edition . Th e author consider s this improvement, du e to Jones, to b e a majo r one . N o Singl e axiom i n th e set thus revise d implie s the existenc e o f a simple close d curv e o r eve n o f a n are. In eac h o f a comparativel y smal l numbe r o f instances , th e nam e o f a mathematician ha s bee n writte n i n conjunetio n wit h a theorem. Ther e ar e vii vin PREFAC1 probably man y case s i n whic h i t coul d hav e bee n don e just a s appropriatel y as in the case s where it has bee n done . Th e author di d no t however car e t o assume the responsibility o f fixing the credit for more than a small proportio n of th e proposition s proved . Thu s th e fac t tha t n o nam e i s attache d t o a particular theore m an d tha t n o reference i s made i n connectio n wit h i t doe s not, b y an y means , necessaril y impl y tha t n o on e other tha n th e autho r ha s stated th e theore m i n questio n an d prove d i t b y a n argumen t tha t woul d apply, wit h littl e o r n o change , o n th e basi s o f th e axiom s employe d i n th e treatment give n here . O n th e othe r hand . th e fac t tha t th e nam e o f a mathematician i s give n i n connectio n wit h a theore m doe s no t necessaril y imply that th e proo f give n i n this boo k i s due to that mathematicia n o r eve n that an y argumen t tha t ha s bee n give n b y hi m woul d b e at al l sufficien t t o prove th e theore m i n questio n o n the basi s o f the axiom s her e employed .