HOMEOMORPHISMS OF STONE-CECH COMPACTIFICATIONS by YING NG B.Sc, Taiwan Normal U n i v e r s i t y , 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 In presenting an advanced the I Library further for this thesis degree shall agree scholarly at the University make that by his representatives. of this written it freely permission purposes thesis for financial of Water es y /£, of British Columbia, gain Columbia shall the requirements f o r reference copying by t h e Head i s understood M gjk-P,MxaXx c ^ The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a of for extensive may b e g r a n t e d It fulfilment available permission. Department Date in partial that of I agree and this n o t be a l l o w e d or that study. thesis o f my D e p a r t m e n t copying for or publication w i t h o u t my ABSTRACT The set o f a l l c o m p a c t i f i c a t i o n s , K(X) of a l o c a l l y compact, non-compact space X form a complete l a t t i c e w i t h |3X, the Stone-Cech c o m p a c t i f i c a t i o n of X as i t s l a r g e s t element, and flX, the one-point c o m p a c t i f i c a t i o n of X as i t s smallest element. F o r any two l o c a l l y compact, non-compact spaces X,Y, the l a t t i c e s K(X), K(Y) are i s o morphic i f and only i f £ X - X and |SY - Y are homeomorphic. pN i s the Stone-Cech c o m p a c t i f i c a t i o n o f the countable i n f i n i t e d i s c r e t e space N. There i s an i s o morphism between the group of a l l homeomorphisms of and the group o f a l l permutations o f N; so |3N has c homeo-r,. * c morphisms. The space N =|3N - N has 2 homeomorphisms. c a r d i n a l i t y of the s e t o f o r b i t s morphisms o f N * onto N i n t o i t s e l f , then * of the group of homeo- c i s 2 . I f f i s a homeomorphism , the s e t o f a l l k - p e r i o d i c of f i s the c l o s u r e of P , n N The i n |3N. of points iii ACKNOWLEDGEMENTS I am deeply indebted to Professor J . V. Whittaker for suggesting the topic of t h i s thesis and f o r rendering invaluable assistance and encouragement throughout the course of my work. I would l i k e to thank Professor T. E. Cramer f o r reading the f i n a l form of this work. I gratefully acknowledge the f i n a n c i a l support of the Department of Mathmatics of the University of B r i t i s h Columbia. iv TABLE OF CONTENTS Page 1 INTRODUCTION CHAPTER 0 : Preliminaries CHAPTER I : A Theorem on homeomorphisms o f j3X - X and 2 j3Y - Y, with X and Y l o c a l l y compact, noncompact Hausdorff spaces 27 Homeomorphisms o f j8N 67 CHAPTER I I I : Homeomorphisms o f N* 73 CHAPTER Homeomorphisms o f 91 CHAPTER II: IV: BIBLIOGRAPHY |SN i n t o |3N 103 INTRODUCTION For any l o c a l l y compact, non-compact Hausdorff space X, at least we have |3X, the Stone-Cech compactification and AX, the one-point compactification. In chapter I, we study K(X), the set of a l l compactifications of X, a l o c a l l y compact, non-compact Hausdorff space, and p a r t i a l l y order i t by defining cx^X ^ oCjX i f there exists a continuous function mapping o^X onto o<jX which leaves the points of X fixed. K(X) i s a complete l a t t i c e under such an ordering. We consider (3X, ^Y, the Stone-Cech compactif ications of X, Y and find out the r e l a t i o n between K(X) and K(Y) when there i s a homeomorphism from |3X - X onto pY - Y. In chapter I I , we study homeomorphisms of p N onto j3N where N i s a countable discrete space. We study the homeomorphisms of N = j3N- N onto N i n chapter III and the homeomorphisms of |3N into |3N 4-- ( including into N ) i n chapter IV. * df Du-Bois-Reymond separability of N The property plays an imp- ortant role i n the proofs of theorems concerning homeomorphisms of N p . onto N and those of ^SN into 2 CHAPTER 0 PRELIMINARIES [4] and [6] are considered as the standard references. Proofs given by these text books w i l l not be repeated here. However, notations, d e f i n i t i o n s , and fundamental results required by or used throughout this thesis are summarized i n this chapter. This chapter i s divided into four sections. In section 1, the fundamentals f o r a topological space are presented. In sections 2,3, and 4, the fundamentals of the spaces (3X, (JN and N are given. 3 Section I 0-1.1 Definitions : (1) C(X) i s the set of a l l continuous functions from the space X to B (reals). (2) 0 (X) i s the set of a l l bounded continuous functions from X to R. (3) Z(f) ={x € X : f ( x ) = 0 } , the zero-set of f , i s the set of a l l elements of X whose image i s 0, where f £ C(X). (4) Any set that i s a zero-set of some function i n C(X) i s called a zero-set i n X. (5) Z(X) ={z(f): f«C(X)} i s the family of a l l zero-sets i n X. 0-1.2 Theorems : (1) Every zero-set i s a G^ and i s closed. (2) Z(X) i s closed under countable intersections. 0-1.3 Definitions : (1) Two subsets A and B of X are completely separated i n X i f there exists a function such that f € C (X) f [A)= {o}, f[B] = {l] and O ^ f ^ l . (2) A subspace S of X i s C -embedded [C-embedded]in X i f every function i n C (S) [C(S)J extended to a function i n C (X) 0-1.4 can be [c(X)] . Theorems : ( l ) Two subsets of X are completely separated i f and only i f they are contained in disjoint zero-sets i n X. (2) Urysohn's extension theorem : A subspace S of X * i s G -embedded i n X i f and only i f any two com-? pletely separated sets i n S are completely separated i n X. 0-1.5 Definitions : (1) A non-empty subfamily J of Z(X) i s called a Z - f i l t e r on X provided that (i) 0 4 1 (ii) if Z l t Z € J.then 2 Z^ 3F; and ( i i i ) i f Z e J , Z* £ Z ( X ) , and Z ' D Z , then z'e J . (2) Every family (8 of zero-sets that has the f i n i t e intersection property i s contained i n a Z - f i l t e r : the smallest such i s the family J of a l l zero-sets containing f i n i t e intersections of members of (3 . We say that © generates the Z - f i l t e r ? . When (6 i t s e l f i s closed under f i n i t e intersection, i t i s called a base f o r J . ( 3 ) By a Z - u l t r a f i l t e r on X i s meant a maximal Zf i l t e r , i . e . one not properly contained i n any other Z - f i l t e r . Thus a Z - u l t r a f i l t e r i s a maximal subfamily of Z(X) with the f i n i t e i n t e r section property. 5 0-1.6 Theorem : If , Q> are d i s t i n c t tnen there e x i s t s EcX 0-1.7 u l t r a f i l t e r s on s . t . E«I2j and X - Ee.Q^. Remark : In a d i s c r e t e space, every set i s a set, so that f i l t e r s and Z-ultrafilters 0-1.8 X, zero- Z-filters; ultrafilters and are the same. Definitions : (1) A space X i s s a i d to he completely r e g u l a r provided that i t i s a Hausdorff space such t h a t , whenever F i s a closed set i n X and xeX letely (2) - F, F and jx} are comp- separated. A c o l l e c t i o n (8 of c l o s e d sets i n X i s a base f o r the d o s e d sets i f every closed set i n X i s an i n t e r s e c t i o n of members of (ft. E q u i v a l e n t l y , (B i s a base of closed sets i f whenever F i s closed and xeX there i s a member of (8 that contains but not (3) - F, F x. A f a m i l y (8 of c l o s e d sets i n X i s a base f o r the c l o s e d s e t s i f the sub- finite unions o f i t s members c o n s t i t u t e a base f o r the c l o s e d s e t s . 0-1.9 Theorems : (1) Every closed set i n a completely regular space i s an intersection of zero-sets. (2) A Hausdorff space i s completely regular i f and only i f the family Z(X) of a l l zero-sets i s a base for the closed sets. (3) (a) In a completely regular space, any two d i s j o i n t closed sets, one of which i s compact, are completely separated. (b) In a completly regular space, every G^ containing a compact set S contains a zero-set containing S. every compact Hence, i s a zero-set. 0-1.10 Theorems : (1) (a) Every subspace of a completely regular space i s completely regular. (b) In a completely regular space, i f f(x)=f(y.) VfeC, then x = y. (c) R and a l l i t s subspaces are completely regular. (2) I f (B i s a base C subbase ] for the closed sets i n X, then { A = X - B : Be©) forms a base I subbase 1 for the open sets i n X. 7 (3) (a) Every closed set F i n a completely regular space i s an intersection of zero-set-neighborhoods of F. (b) Every neighborhood of a point i n a completely regular space contains a zero-set neighborhood of the point. 0-1.11 Definition : Let X be a completely regular space. A point p € X i s said to be a cluster point of a Z - f i l t e r 5 i f every neighborhood of p meets every member of J- , i . e . p i s a cluster point of J i f and only if p e ^ e Z - f i l t e r $ i s said to converge to the l i m i t p i f every neighborhood of p contains a member of J . 0-1.12 Theorems : 3F i s a Z - f i l t e r on a completely regular space X, p € X: (1) (a) I f J" converges to p, then p i s a cluster point of 5 . (b) A Z - u l t r a f i l t e r 3 converges to p i f and only i f ? contains the Z - f i l t e r of a l l zero-set neighborhoods of p. (c) I f p i s a cluster point of J .then at least one Z - u l t r a f i l t e r containing J converges to p. (2) The family of a l l zero-sets containing a given point p i s denoted by Ap : (a) p i s a cluster point of a Z - f i l t e r J i f and only i f Jcz Ap. (b) Ap i s the unique Z - u l t r a f i l t e r converging to p. (c) D i s t i n c t Z - u l t r a f l i t e r s cannot have a common cluster point. (d) I f J i s a Z - f i l t e r converging to p, then Ap i s the unique Z - u l t r a f i l t e r containing J . 0-1.13 D e f i n i t i o n : A hausdorff space i s said to be l o c a l l y compact provided that every point has a compact neighborhood: i t follows that every neighborhood of a point contains a compact neighborhood of the point. 0-1.14 Theorem : Let X be a subspace of a Hausdorff space T. (a) I f T i s l o c a l l y compact, and X i s open i n T, then X i s l o c a l l y compact. (b) I f X i s dense i n T, then every compact neighborhood i n X of a point p f X i s a neighborhood i n T of p. (c) I f X i s dense i n T and p i s an isolated 9 point of X, then p i s isolated i n T. (d) I f X i s l o c a l l y compact and dense i n T, then X i s open i n T. 0-1 .-15 D e f i n i t i o n : A point p i n a topological space X i s a P-point of X i f every countable intersection of neighborhoods of p contains a neighborhood of p. 0-1.16 Theorems : (1) I f X i s completely regular, then p i s a P-point of X i f and only i f f o r every f e C(X), there exists a neighborhood U of p such that f i s constant on U. (2) Every countable set of P-points i n a Hausdorff space i s discrete. 0-1.17 Definitions : A family of subsets of a space X i s a ring i f i t i s closed with respect to f i n i t e unions, f i n i t e intersections and complementations. 0-1.18 Theorem : Each countable family of sets i s contained i n a countable r i n g . 0-1.19 D e f i n i t i o n : A topological space X i s homogeneous i f to every pair of points p and q of X, there exixts at least one homeomorphism of X which carries p to q. 10 Section II Throughout t h i s section, a l l given spaces are assumed to he completely regular. 0-2.1 Definitions : (1) By a compactification of a space X, we mean a compact space i n which X i s dense. (2) A Z - f i l t e r i s free or fixed according as the intersection of a l l i t s members i s empty or non-empty. 0-2.2 Lemma : A zero-set Z i s compact i f and only i f i t belongs to no free Z - f i l t e r . 0-2.3 Theorems : Suppose X i s dense i n a space T. (1) I f J i s a Z - f i l t e r on X, then p e T i s a cluster point of J provided P«Z& (2) C 1 T that Z I f Z i s a zero-set i n X, and p e Cl^Z, then at least one Z - u l t r a f i l t e r on X contains Z and converges to p. (3) Every point of T i s the l i m i t of at least one Z - u l t r a f i l t e r on X. 0-2.4 Theorem : Every completely regular space X has a Stonefiech compactification j3X with the following 11 equivalent properties .: (1) (Stone) Every continuous mappingz from X into any compact space Y has a continuous extension Z (2) from pX into Y. (Stone-Cech) Every function f i n C * ( X ) has an extension to a function (3) i n C(|3X). (dech) Any two d i s j o i n t zero-sets i n X have d i s j o i n t closures i n p X . and Z 2 i n X , (4) For any two zero-sets (5) D i s t i n c t Z - u l t r a f i l t e r s on X have d i s t i n c t l i m i t s i n |3X. * (6) I f X i s dense and C -embedded i n T, then X.crT< (7) I f X i s dense and C -embedded i n T, then |3T = pX. Furthermore, p X i s unique, i n the following sense: i f a compactification T of X s a t i s f i e s any one of the l i s t e d conditions, then there exists a homeomorphism of p x onto T that leaves X pointwise fixed. 0 - 2 . 5 Construction of p X : (a) The points of |3X are defined to be the l i m i t points of Z - u l t r a f i l t e r s on X . So the family of a l l Z - u l t r a f i l t e r s on X i s written ( A ) p where A p point p. p e i s the Z - u l t r a f i l t e r on X with l i m i t ^ 12 (b) pe[3X : Z <£ A j Define Z p forZeZ(X). J3X i s made into a topological space by taking the family of a l l sets 2 as a base f o r the closed sets. C1. Z T (c) = 2 . p e C l Z i f and only i f Z e A . p flT Every point of |3X i s the l i m i t of a z - u l t r a f i l t e r on X. So there i s a natural correspondence of elements of ^X and Z - u l t r a f i l t e r s on X. Corresponding to 2 = {p e J3X : Z e A } p we have Z = { n : Z £ Ii, £2 i s a Z - u l t r a f i l t e r on X} for Ze Z(X), when we consider elements of (SX as Z - u l t r a f i l t e r s on X. 0-2.6 Theorems : S i s a subspace of X, (1) S i s C*-embedded i n X i f and only i f ci s=ps (2) Every compact set i n X i s C -embedded i n X. (3) I f S i s open-closed i n X, then C1., S and Y Clpy^X - S) are complementary open sets i n (4) An isolated point of X i s isolated i n X i s open i n compact. j}X, |3X. and |3X i f and only i f X i s l o c a l l y 0-2.7 Theorem : For any i n f i n i t e discrete space Where IXI i s the cardinality of X. 14 Section I I I The set N of a l l positive integers i s chosen as a countable discrete space. Since N i s l o c a l l y compact, N i s open i n p N . Every point of N i s an isolated point of p N . These are the only isolated points of p N , since N i s dense i n p N . Hence every f i n i t e subset of N i s open-closed in pN. Since every subset of N i s open-closed i n N and i s also a zero-set so the closure i n p N of every subset of N i s open i n p N . 0-3.1 .Theorem : Every open-closed set i n |6N i s of the form Clp^A f o r some A c l . So there are c open-closed sets i n pK, where c i s the continuum cardinality. Proof :Suppose Sep i s open-closed i n p u , then B = pN - A" i s also open-closed i n jSN. Let A = SDN, ClpjjA B = B/1H. Clearly I = C l ^ A , i f not, A" (by d e f i n i t i o n , 01 ^k i s the smallest closed set i n p N that contains A). Let A' = A - C l ^ .•A which i s non-empty open-closed i n ^QN. Since N i s dense i n 0-3.2 pN, A'riN £ 0. This i s a contradiction. Remark : It follows from the construction of p x that 1 the open-closed s e t s of AN form a base f o r the c l o s e d sets of AN. Since the complement of every open-closed set of 0N i s a l s o open-closed i n -3N, so they a l s o form a base f o r the open sets of AN. (by Theorem 0-1.10(2)). 5 16 Section IV # Now we consider the subspace N = p N - N of j@N. Since (3N i s compact and N i s open i n j8N, so N i n pN and therefore N i s closed i s compact perfect. For Ac:N, define A' = G l ^ A - N = C l ^ A fiN*. Clearly, A i s f i n i t e i f and only i f A'= 0, and A ' c B ' i f and only i f A - B i s f i n i t e ; so A' = B ( A - B ) U ( B - A ) i s finite. i f and only i f 1 It follows that there are c d i s t i n c t sets A'.(c i s the continuum cardinality.) Since { C l ^ A : A c N forms a basis f o r the open sets of (JN and also a basis f o r the closed sets of |3N, so does {A = ClpjjArtN*: A C N ( f o r N* . 1 0-4*1 Theorem : * Every open-closed subset of N i s of the form A' = C l ^ A - N = C l ^ A H N * f o r some A c N . Proof :Suppose W i s open-closed i n I ,V = N i s also open-closed i n N . Since p N , N -W are compact, then normal, there exists a continuous f : p N -*• CO,M s.t. flW] = (Oj , fEVl ={1} . Let A = { n e N : f (n) < -j-}, B = |n € N : C= f (n) > £}, N - A - B = f n e N : f (n) = i ) . Since jx €pN : f(x) = ±) nN* = 0 and {xe |SN : f(x) = *} i s closed i n |?N, 17 we have C l p CflN = 0 . .'. C i s f i n i t e ; and ¥ = A* = C l ^ A fiN*. 0-4.2 Theorem : The intersection of any countable family of open subsets of N i s either empty or contains a non-empty open set. Proof :Let of N {G^} be a countable family of open subsets whose intersection contains a point a. Since [A' = C l ^ A ^ N * : A c N J forms a basis f o r the open sets, there are i n f i n i t e sets A^c N s.t. aeA|c:G i V i e N. The intersection of any f i n i t e c o l l e c t i o n of the sets A^ i s non-empty and open. For F, any f i n i t e c o l l e c t i o n of the sets A^, there corresponds F*, a f i n i t e , c o l l e c t i o n of the sets A^. Pi A ! ^ 0 AjeF* Since and x A ^ V i OT-»A. A = C 1 N A9F 1 A F (Theorem 0-2.4(4)), i s i n f i n i t e . Thus there exists an increasing sequence of integers n^ s . t . so chosen, then A - A I i s f i n i t e f o r each i € N 18 because zi^A-^ whenever j ^ i , A - A. c f i i | , ... n ^ ] ; so A'<-=A£ " A c ieN A i VieN. ieN i u A is infinite, .*. A» ji 0 and A' i s open-closed i n N• . 0-4.3 Definitions : If S i s a set p a r t i a l l y ordered by -< , e, h^, l ^ . h, e.j , e2> •••••» © > n , e.|, C2» ••••»c , , h^, .... are i n S, then n S i s said to have the properties : (1) Simplest separability , i f f o r each e< h, there exists a g e S s.t. e<g-<h; (2) Separability of Cantor, i f f o r any set e.j< ... <e 4 ^ h of type n there exists age.S s.t. w+ 1 , e -<g-<h Vn £ N; n and (3) Separability of Du-Bois-Reymond, i f f o r each set e < e^< ....-< e < .... < h < .. 1 .. < hp < h.j n of type w + c , c , ...s.t. 1 2 h ^ c n , and any set and ± e Q for any n and i e N , there exists a g e S s.t. e <g<h n n any i £ N. V n €N, and c^ ± g ^ o for ± 19 0-4.4 Theorem : Let LN be the f a m i l y of a l l open-closed -si- # subsets of N , a b a s i s of N , with the p a r t i a l order of s e t i n c l u s i o n , LN has the p r o p e r t i e s of simplest s e p a r a b i l i t y , s e p a r a b i l i t y of Cantor and s e p a r a b i l i t y o f Du-Bois-Reymond. Proof :* (1; LN i s simplest separable : Let A» f B» Since N i s perfect, H y ^ Z e l B ' - r A ' ( B e LN*, 3 y e ( B' - A' ) € LN*, - A' - Z ) i s an open but not c l o s e d 1 neighborhood of y. .\ 3 0 j t C ' e LN* s . t . y € C'c B - A' - Z f A'^C'UA'^t 12) B' - A* ) U A ' = B' LN i s Cantor separable : Let Aj g A£ £ . . . ^ £ ... fF B' B« , A| €LN* Put 1 Clearly ^ where V i eK, C| =• N* - A | D Set ) VieN and = N* - B*. 0^ e LN* VieN, D' e LN*, and * '''' * n ¥ E£ = G* - D' . ± Ej_ e M * V i e N, and * * By Theorem 0-4.2, we have j^EJj^ c o n t a i n i n g an H' s . t . H* Let K = N* - 1 LN*, and 6 ( H'U D ), 1 then * LN i s Du-Bois-Reymond separable : Let Aj_ and B| be i n LN* - 1 4: • • • • - ° * •••• ^ D i = ii - ^ B A i ' B 1 1 B < B 1 2 with 0 A. A N * N s.t. A -| • ( BN i * ' ^ subsets of N, V i eN, and 1 B 1 1 1 1 < + n A£ = C1^ with Vi eN n N 6 <B < ^ A< n ....-< A -< A n 2 1 •< a p a r t i a l order d e f i n e d by A< B if and only i f A - B i s f i n i t e and B - A is infinite. A'cfcC! n T x Let C*, G^, .... and C^B^ G i = G1 be i n LN V i , n e N, |3H j _ ^ * » G N c i an i n f i n i t e subset o f N. i . e . °i* n ^ A A,, - C. and n 1 all i , n 6 N. C i ^ n B o r C. - B_ are i n f i n i t e f o r i n s 21 Now construct E^,K^ by induction as follows : H K = U j H B,) \J{V ] where p 1 U 1 = Unl where *n °i ~ i 6 1 C because U f P ^ I and 1 - H 1 U {P )r>C 1 11 - B 1 infinite. 1 Suppose H , H , ...., H _ , 1 2 n 1 2 n = k.(\ k f\ finite Vi = n-1. 1 ( which i s f i n i t e , and n 2 H ={m I flB n 1 f)k . 0 n 2 be an integer greater than any member n of K (J K U .... U K _ L , K , ...., 1 are constructed, with Let t - ^ 1 H can be chosen i n C^H CB £A 1 1 n : m >^ } U n { p^.ito } where p. i n e Ln - C. l and p. m > t.n. f o r i = n. The choice of p i s possible since A - CL i n n i s i n f i n i t e and A - L = A - A, 0 A H n n n 1 2 n f\k n 0 iWl n " i^ (A since A n < A^, , < . . . . • < n—1 A 2 0 < A „ I , A A ^ - n i s A . i f i n i t e is f i n i t e f o r a l l i £ n. H n has the following properties : (i) H n i s infinite : nB L n = n nk n— A . 1 = ( A n A r\ B 0 2 1 nB n )nu 2 n n n B ) n.... n For 1 = 1 , 2 , ....,n A^ i s i n f i n i t e B N - A ± i s finite ( V B -< A ) N ± n(A n n B N ) 22 13 i s i n f i n i t e n .'. (A f\ B ) i s i n f i n i t e , and n i n iQl ^ i 0 2 ^ ' = G n ( 0-4.1, 0-4.2) T h e o r e m s n with G = S)^ {A f) B ) n ± infinite n (Theorem 0-2.4(4)). H D G - }m : m < t ) i s i n f i n i t e . (ii) H . - B i s f i n i t e V i £ n: l n n n n ( Pin * ) n H n C n L H nB n U : n {Pin C B U 1 n : .*. Hn - B„ n is finite •'• i "n - i " V " <i - V H B C H B which i s f i n i t e V i ^ n s i n c e B (Iii) B - B ± i s f i n i t e V i £ n. n - H i s finite : n n n " n ^ n B H C B " ^ L n n B n ) U ( m £ L n n B n : m ^ t ] and n n" • n n > =n " < 1 ^ • • • = iW,< n - V B L nB B A B B„ - A. i s f i n i t e n x V i e N. .*. B^ - H i s f i n i t e , n n (iV) H - A n Hn B n .*. C k B i s finite n U {Pin Vk € N : : i - A is finite k H - A n k ~ ) n a n d Vk € N i s finite Vk e IT. > 23 (V) H < A n H n Vi€ N i ^ f in n C B P * ) : i n P> - A. f i n i t e V i e N n 1 .'. H - A, f i n i t e n 1 Vi e N A. - B„ i s i n f i n i t e l n V i eN A. - H i s i n f i n i t e x n .'. H < A, n l (Vi) H C k n n V i eN. Vi = n ± H c L V i e N. = A n 1 A n D .... n 2 A ^ k n Vi ^ n ± Define K = { « - J ^ : q i n > V i = nj. The choice of q ^ i s possible because n C - B i n f i n i t e V i , k e l , and ^ \ - B n q ± n 0 i n k i n i s f i n i t e by property (ii) of H . R C. i -n B„ i s i n f i n i t e V i , n eN. n C. - , U H, i s i n f i n i t e , l n=1 h Put 5.Hn D = n=1 For any fixed k we have D = P^U Q P Q = H U H U k k 1 = H k U H 2 + 1 U H k + 2 We have shown that (property (iii) of H .'. B k - P k n k where k U B - H is finite n n ) is finite 24 B - D k i s finite " k = ^+1 k+2 D P U H '' *' ^ ^+1 U infinite. .*. Also B 4D Q ^A k V \ k + 1 = M \+2 E U — U • C ( by property (Vi) o f H •'• Q .. •'• is k 1 + ) n k ~ k+1 = $ A P k - H j P D A M k+1 < A - A k V k + 1 - k+1 A j € N property (Vi) ) ( is finite. = < k- k+1 P A ) U ( Q k ~ W finite. A k + 1 <A k given A k - A k + 1 i s infinite A k - D = A k - ( DfU => k " k-M " A A ( )- (D - A k + 1 D " k 1 A } i k + 1 ) S + infinite. D <A k Hence we have B, -< B < ... < B < ... <J>< ... <A 1 2 n < ... < k. n 1 25 For any fixed CL, there are i n f i n i t e l y many p. « D , p. 4 C., so D - C. i s i n f i n i t e . The choice of the number q 0 . - D i s infinite. i n shows that Thus D* has no set x inclusion r e l a t i o n with 0 / V i e N. 0-4.5 Definition : In 0 - 4 . 4 ( 3 ) , M'0 B | = 0 if 3 M' € LN* s.t. M' (\ Aj_ £ V i e N , then we can have D' d i s j o i n t with M' or not by setting D* as the union of the D' described above and Q' where Q'e LN and Q'c M' 0 i n addition ( By Theorem 0-4.2 ). I f P' = M» H J^^ A | i s open-closed i n N , we may choose D' s.t. P ' C D* by choosing D' as the union of the D 0-4.4 and P'. 1 described i n Theorem Let this property be called the extended Du-Bois-Reymond separability. 0-4.6 0, Definition : We may say that a zero-dimensional space X ( one with a basis of open-closed sets ) has the properties of simplest separability, Cantor separability or Du-Bois-Reymond separability i f the open-closed set basis of the space with the p a r t i a l order of set inclusion has the corresponding property. 26 0-4.7 Remark : For a zero dimensional compact space, simplest separability i s equivalent to perfect. ( 4= i s c l e a r l y shown i n the proof of Theorem 0-4.4(1). point, 0-4.8 •=->• i s true, f o r i f x i s an isolated 3 no B s.t. 0 fi B ^ {x}.) Remark : N i s simplest separable, Cantor separable and Du-Bois-Reymond separable. CHAPTER I A THEOREM OH HOMEOMORPHISMS OF pX - X AND <3Y - Y ' WITH X AND Y LOCALLY-COMPACT NON-COMPACT HAUSDORFF SPACES In t h i s chapter, a l l g i v e n spaces and a l l c o m p a c t i f i c a t i o n s are Hausdorff and a l l g i v e n l o c a l l y compact spaces are non-compact. 28 Section I : In t h i s s e c t i o n we study the set of a l l compactifications of a l o c a l l y compact, non-compact, Hausdorff space. For b r e v i t y ' s sake, i n t h i s s e c t i o n we f r e q u e n t l y use the a l t e r n a t i v e d e f i n i t i o n of compactification : a compactification is a pair ( o<X, h ) c o n s i s t i n g of a compact Hausdorff o(X and a homeomorphism h oCX. Two of a t o p o l o g i c a l space X compactifications of X onto a dense subset of c<X, there e x i s t a homeomorphism (p of ifh^x) = h (x) y Vx TX of X are equal i f c*X to TX s u c h that eX. I f ( (XX, h^) i s a compactif i c a t i o n of X, c<X - h CXI = 3(X, d X r e l a t i v e to 1-1.1 oCX) i s the boundary of <*X. Definition : l e t X be a t o p o l o g i c a l space, T a compact space. A f a m i l y J(X, T) of continuous f u n c t i o n s i n t o T i s completely r e g u l a r i f f o r every FcrX and x f(x ) Q Q 4 F, there e x i s t s f e ^ C X , 4 fTFT . from X closed T) s u c h that 29 1-1.2 Theorem : Let X he Hausdorff, T compact,? a completelyregular family of continuous functions of X into T. There exists one and only one compactification JX of X associated with the family J with the following properties : (a) Every fe 1 can be extended to a continuous f of JX into T. Yx eX 3 . C f(hj(x)) = f(x) (b) I f y', y" e 7X, y» £ y" V l i M f e J then there exists s.t. Proof :(1) There exists such a compactification of X : Consider the compact Hausdorff space * fejF f x where T^ = T whenever f e J , and the mapping q : X —*• E defined by 7i (q(x)) = f(x) f Vx € X where TC^ i s the projection 7l : f E If f e J, then T . f 71^,* q i s continuous and, so i s q. Since X i s Hausdorff, J i s completely regular, f o r any x', x"€ X, x there exists f € 7 s.t. f ( x ) £ f ( x " ) , hence f 1 £ x", *n UU")) Tt (qU')) f f i . e . q i s one-to-one. Let G be open i n X and X e G. Then X - G i s q closed and x Q 4 X - G. so there e x i s t s f(x ) Q J i s completely regular, f€?s.t. * fCX - G] , .'. there e x i s t s Y , Q an open neighborhood of f(x ) s.t. Q V„ n f CX - G] = 0 . o f i s continuous, .'. x ~ ^ ( V ) f 0 q^) i s open i n E, and we have e q(X)n7U "" (V )cq(G) 1 f 0 .*. qC^Q ) 6 q(G) and qCG3 i s open i n q l X l . Let ?X = q[X] and h^: X —* ?X be d e f i n e d by : h j ( x ) = q(x) Clearly, V x eX. (3FX, h^) i s a compactif i c a t i o n o f X. We have the diagram : where f i s r e s t r i c t e d to ?X. If y', y"e 7X, y' £ y" there exists f € 3 s. t . TTfCy') ^* (y") f .V f ( y ' ) ^ f(y") So (JX, hj) i s a compactification of X that s a t i s f i e s (a) and (b) above. (2) The compactification (JX,h ) i s unique: ? Let (<*X, h^) be another compactif i c a t i o n of X with properties (a) and (b). Define if» : o ( X - » ? X by «P(y) = (f ' ( y ) ) f € j , Vy o(X € where f* i s the continuous extension of f to <*X. Then we have the commutative diagram: X 5X The function y i s continuous since every f ' i s continuous, <f i s one-to-one, by property (b) of I f l f e ? . f((XX) i s compact, Y(^X) -=> hy(X), because (pChrfCx)) = hy(x) of 4> ; 32 f o r a l l x e X by d e f i n i t i o n and hy(X) i s dense i n J X . Therefore f ( dX) = Therefore f i s a homeomorphism of ?X, with J(X) o(X onto <f(h^(x)) = hj(x) . Therefore tfX = JX. 1-1.3 Corollary : A space i s compactifiable ( has a compactification ) i f and only i f i t i s completely regular. In the rest of this chapter a l l given spaces are completely regular. 1-1.4 Definition : A compact Hausdorff space T i s practicable i f there exists a continuous function which i s not constant from [0,11 1-1.5 into T. Theorem : Every compactification of a compactifiable space X i s associated with a completely regular family of continuous functions from X into a given practicable space T. Proof :Let ( o<X, h,) be a compactif i c a t i o n of X. Since T i s practicable, there exists a continuous if which i s not constant such that 33 if maps CO, 11 t o T. W i t h o u t l o s s o f g e n e r a l i t y , we may assume «P(0) ?t (f (1 ) . We i d e n t i f y h ^ X ) w i t h X. Let 1 he a c l o s e d s e t i n X a n d x e X - F „ , o 1 then there x- exists F e c<X - F c l o s e d i n e<X s u c h 2 a n d F^ = F , , n X. 2 then normal, there e x i s t s a continuous o(X t o 1 0 , f t h a t maps Since that o<X i s c o m p a c t , function 11 s u c h t h a t f(x ) = 0 o f CF Therefore Since I 2 = {1} f(0) = <f ( f ( x ) )^ <f* ( f CF1 ) = Q (f(l) i n T. T i s Hausdorff, no) that i s ^ wry <M ( o^ ^ <f f ( x ) ^ f x Q Therefore f ^ l x 1 S vTfCFpT <ff CF 1 . 2 a f u n c t i o n from X i n t o T such that (a) <Pfl (x ) (b) ^ f l ^ x 0 4 ^f l (F x 1 ) and has a continuous e x t e n s i o n t o (*X. Now l e t 3^ be t h e s e t o f a l l c o n t i n u o u s of X i n t o T with continuous extension \ i s uniquely a completely d e t e r m i n e d by regular family. f o r y', y" i n dX, fy' *f y " . (XX and T, functions t o <XX. Then and i s I t i s also clear that y", t h e r e exists f e % s o that 34 1-1.6 Theorem : Let o<X be a compactif i c a t i o n of X, T practicable, be the completely regular family of continuous f u n c t i o n s a s s o c i a t e d with o<X = Then X. Proof :- The compactif i c a t i o n ions (a) and J family U Therefore 1-1.7 o<X. (?<X s a t i s f i e s both c o n d i t - (b) of Theorem 1-1.2 r e l a t i v e to the . the two c o m p a c t i f i c a t i o n s are equal. Definition : Let X be c o m p a c t i f i a b l e and K(X) be the set of a l l c o m p a c t i f i c a t i o n s of X. For two <XX, YX of K(X), c<X = yX i f there e x i s t s a continuous f u n c t i o n ¥ from tfX onto »f(h (x)) = h ^ x ) y f o r a l l x g X. h IX] are dense subsets elements of (A X and oCX such that Since h< CXI TX r e s p e c t i v e l y , y and Thus ip i s continuous, ip i s u n i q u e l y so <f EoCXI i s compact. determined. C l e a r l y the r e l a t i o n =" i s a p a r t i a l i n K(X). From theorem 0-2.4, we have l a r g e s t element of K(X). and order |3X the 35 1-1.8 Lemma : YX be two Let X be compactifiable, <?(X, compact i f icat ions of X such that <XX = Y X . T X to o<X induced Then the function if that maps by the r e l a t i o n =" s a t i s f i e s the condition. f c 9 u , r x ) a = a ( x , <-<x) <f C T X - X i = o(X - X. that i s Proof :(1) Clearly if E Y X - X I => cCX - X, since <p i s onto and i t leaves X invariant. f t tX - (2) XI c oCX - X. If ye YX - X such that clearly <f(y) = x *f (x ) = x . Q Q f o r some x € X c o(X and Q Since x Q cp(y) ^ <*X - X, then open neighborhoods Q ^ y i n Y X , we can choose V of x Q and U of y i n YX V n U = 0; then Yf\ X i s open i n X, such that and there exists open neighborhood i n o(X such that V O X = continuous, f o r hood V Y x 1 Since f i s i n o(X there i s a neighborT 1 i n <*X. On the other hand, X i s dense i n Y X , there exists x^ e clearly Q of y such that V C U i n Y X and if I V ] C V Y X. of X x^ XOYT'^U then ^ V because U f l V = 0, and = if (x^ ) e V c o(X. Therefore x ^ ^ H X 1 This i s a contradiction. = VOXcV. 36 1-1 ..3 Theorem : L e t X be a t o p o l o g i c a l J T two c o m p l e t e l y r e g u l a r functions where space, T p r a c t i c a b l e , f a m i l i e s of continuous If of X i n t o T. ^ c ? r , then dX ^ YX <<X, JX a r e t h e c o m p a c t i f i c a t i o n s o f X ciated with J,» asso- 3* , 3y r e s p e c t i v e l y . the families P r o o f :- Therefore every f e ^ continuous f of *X. c a n be e x t e n d e d L e t i f , a map f r o m to a KX t o o(X be d e f i n e d b y : <p(y) = Clearly diagram so . means t h e f o l l o w i n g i s commutative : Evidently or f € 3 P - «flhj(x)l = h^Cx) t h a t x <XX, {^y)J 1 Y x u» i s c o n t i n u o u s , a n d T J X ] i s compact i n «pEh (x)I i s d e n s e i n o(X; t h e r e f o r e T ^ i s a continuous function o(x ^ jrx. from o(X = YX o n t o o(X. (? I I X J 37 1-1.10 Theorem : Let o(X ^ p(X, fX be two compactif ications of X. n Then i f and. only i f every continuous function of X into T with continuous extension to -XX into T can be extended to a continuous function from fX into T, f o r T practicable. Proof :(1) Necessity : <<X ± XX ; by d e f i n i t i o n , there exists a continuous function <f : HX onto o<X. Let f be a continuous function from X into T with f^ i t s extension to <*X. Then f ^ - f i s a continuous extension of f to XX. (2) Sufficiency : Let , J be the two y completely regular families of continuous functions associated with By hypothesis and -XX, ?X respectively. c. 3 y ; and by theorems 1-1.6 1-1.9. %X = -XX and JyX = TX and hence *< X ^ YX. 38 1-1.11 Theorem : Let K(X) be the set of a l l compactifications of a given space X. Then every non-void subset of K(X) has a supremum : Proof :Let {c<X} be a nonvoid subset of K(X), T practicable, the completely regular family of continuous functions of X into T associated with <AX. Let By Theorem 1 - 1 . 6 , 3- = JJ and ft ^X = o(X. ST-X be the compactif i c a t i o n of X associated with J. By Theorem 1-1.9, <*X = JX. If 9 X i s a compactification of X such that o(X * 9 X f o r a l l <* e f\ , then by Theorem 1-1.10, every f e ? can be extended to a continuous function of ©X. Therefore J a 3 Therefore 3X £ JX Therefore i X = sup <AX. Q Q = 9 X .. 39 1-1.12 Theorem : I f K(X) i s the set of a l l compactifications of a l o c a l l y compact space X, then every non-void subset of K(X) has an infemum. Proof:Since X i s l o c a l l y compact, (ZX, the one-point compactification of X i s obviously the smallest element of K ( X ) . Let then {o(X} ^ e A c K(X). ax i s the infemum. If aX€{ofX) 4 If ax o ( £ A {oCX}^ and i t does not contain a minimum (else i t i s t r i v i a l ) then by Theorem 1-1.11, j }fX : yX < o(X V *<€ A} has a supremum Y^X i s the infemum of X^X. Then |<^X) ^ « e A 1-1.13 Corollary : If X i s l o c a l l y compact then K(X), the set of a l l compactifications of X forms a complete lattice. 40 Section 2 : In the l a s t section we have defined on K ( X ) , the set of a l l compactifications of X the p a r t i a l order of =". Por any compactif i c a t i o n eKX of X and / 3 X , the Stone-dech compactification of X , we have a continuous function f^ of px onto o<X which leaves X invariant. Clearly f^ i s unique, and this function w i l l be referred to as the 0function of i n t h i s section. erfX By Lemma 1 - 1 . 8 , it is obvious that the p-function f ^ induces a decomposition of | 3 X - X into a family 7 (e<X) empty closed subsets where In t h i s section, ^(t?<X) of mutually d i s j o i n t non- 7(c*X) = { f~^ (p) : p e c<X - XL w i l l be referred to as the /5-family of e < X . 1-2.1 Lemma : c < X , T X be two compactif ications of X . Let Then oCX = T X i f and only i f each set i n J ( y X ) i s a subset of a set i n 3( o < X ) . Proof : ( 1 ) Necessity : I f o<X = y X , then by d e f i n i t i o n , there exists a continuous function h from f X onto o<X such that h(x) = x fom a l l xeX. Let f ^ and f r be the ft-functions of o(. X and T X respectively. h . f (x) = h r So, h« f y (x) = x = f ^ x ) Then for a l l x e X . agrees with f^ on X , a dense subset 41 px. of Since any two continuous f u n c t i o n s from an a r b i t r a r y space i n t o a Hausdorff space must be i d e n t i c a l i f they agree on a dense subspace, thus h»f y = f^ . Now l e t A be any s e t i n J ( JX), then A = f j " ( p ) f o r 1 some p e 7 X - X. Let q e A, then f (q,)= p r and h(p) = h * f (q) = f„<(q) e oCX. So we have r qe f j ( h ( p ) ) and hence A c r f J ( h ( p ) ) a member 1 1 3(o(X). of ( 2 ) Sufficiency of : Let f ^ , f r p-functions be the c<X, XX r e s p e c t i v e l y , p be a point i n YX - X. By hypothesis, there e x i s t s a unique point q i n X X - X so that f 7 Define a mapping h from and h(x) = x £ 1 Y h(f U)).= and T f^U) = t, Suppose A e pX - X, f , 7 ( t ) f o r some t e *tX - X 1 t . Moreover, A € f ~ therefore (f,(A)) c f K we have f^ = h o f . 1 h o f ( f t ) = f^U) h « f ( x ) = h(x) = x = f ( x ) r 1 Y X to <AX by h(p) = q f o r a l l x € X. then f 7 ( f ( i ) ) f,7 (q). (p) S 1 f " (t), and f o r a l l x e X. Hence C l e a r l y h i s onto. Let K be c l o s e d i n <XX, then f ~ ^ (K) i s closed in px, so i s compact; and t h e r e f o r e i s compact, then c l o s e d i n YX. and h " 1 = f ' * f y closed i n YX. have P<X ^ XX. 1 1 fyCf^ (K)) Since f ^ = h»f j< we have h ' ( K ) = f ( f ~ ( K ) ) 1 1 r Thus h i s continuous. So we 42 1-2.2 Remark : Lemma 1-2.1 implies that a compactification i s uniquely determined' by i t s p-family. 1-2.3 Lemma : Let X be l o c a l l y compact, o(X a compactif i c a t i o n of X, , K g , . . . , K J J be N mutually d i s j o i n t non- empty closed subsetsoof c<X - X. Choose N d i s t i n c t points q^ , q ».. • ,q not i n o<X and define a mapping 2 h from N o<X onto by h(p) = p for p e K^. TX = C <*X - J ^ I L I U {q : 1=1, ,NJ ± f o r a l l pe MX - J ^ K ^ and h(p) = q ± Let TX have the quotient topology induced by h,that i s the largest topology on TX such that h i s continuous. Then TX i s a compacti- f i c a t i o n of X. Proof :Since o<X i s compact, TX has the quotient topology induced by h and hCcrfX] = dense i n <XX, we have TX compact and X dense i n TX. Now we have to prove that Let G = (XX - jJLjK^ • each TX, and X i s i s closed i n y x . TX i s Hausdorff: G i s open i n e(X, and G= TX - { q ^ i=1 ,... ,N) so i t i s s u f f i c i e n t to consider the following three cases f o r d i s t i n c t points p and q of to prove that YX i s Hausdorff. TX i n order 43 (1 ) P = .q and q = q m (2) p € G and q = q n m (3) p, qeG. Case (1) ,:•Since each i s closed i n ©<X, which i s compact, there exist d i s j o i n t open subsets U and U of o<X such that m n K c U , K c U , U n K. = 0 f o r m m n n m l i ^ m and U H K = 0 f o r i ^ n. m n ± Let U* m = h(Um ) Both U.. and U m n v since U and U* n = h(U n ). are open i n YX x = h (U*), _1 m U = h" (U*) and 1 n YX has the quotient topology induced by h. Clearly U m and U are d i s j o i n t neighborhoods of p and q respectively. Case ( 2 ) : There exist d i s j o i n t open subsets U and V of c<X such that p e TJ, K G V, ' m ' UHL = 0 for i £ m. Clearly U f o r a l l i and V 0 K = ± 0 Let U*= h(U) and V*= h(V). and V are d i s j o i n t neighbor- hoods of p and q respectively. Case (3) : G i s a subspace of o<X. There exist d i s j o i n t open subsets U and V of G containing p and q respectively. G is open i n o<X, so U and V are open i n p<X. Since h(U) = U and h(V) = V, 44 U and V.are open i n TX i s a compactification of X . Therefore 1-2.4 YX. Lemma : Let X he l o c a l l y compact and l e t «XX be a compactification of X with Suppose K , K 1 2 (3-family J(cAX). 6 ?(*X), let J* =(? (*X) - 4 K , K })U [^U K \ . 1 2 2 T X of Then there exists a unique compactification X with J ( r x ) = J* . Proof : Let f^ be the p -function of <X.X, q^ be the point i n dX- X such that Let K = f j (q ) i=1,2 1 ± i Y X = ( « < X - ( q ^ q } ) U { A} where A i s 2 YX any point not i n o(X. Let h that maps o(X to be defined by h(p) = p f o r a l l p € c<X - {q ,q }, h-Cq^) = A 1 i = 1 , 2 and l e t Y X have the quotient topology induced by h. Then YX is a compactification of X ( by Lemma 1 - 2 . 3 ) . mapping from for a l l x e X . The (3X to Y X i s a continuous function h'f^ mapping j3X onto Y X such that h'f^ (x)=h(x)=x Since the p - f u n c t i o n of a compacti= f i c a t i o n i s unique, h ^ must be the p-function 0 YX, so 7 ( 2 YX)= The compactification of J*. Y X i s unique since any comp- a c t i f i c a t i o n i s uniquely determined by i t s p - f a m i l y . 45 1-2.5 Lemma : Let X be l o c a l l y compact, and {E^: i=1,....,n} be a f i n i t e family of mutually d i s j o i n t , non-empty closed subsets of px - X. Then there exists a unique compactification TX of X such that J ( TX) consists of a l l the sets K^ together with a l l singletons jp} where p e ( pX - X)- . Proof :This i s a special case of Lemma 1-2.3 with c*X = |5X. 1-2.6 Definition: The unique compactification w i l l be denoted by p((X; this section. fication , TX of Lemma.1-2.5 K ) throughout n In case that there exists a compacti— SX of X such that J ( SX) consists of an i n f i n i t e number of closed sets K^ of p x together with a l l singletons we denote 1-2.7 5 x by {p} where p e(|3X - X ) - ^^K^ > o<(X; , Kg,....). Lemma : Let X be l o c a l l y compact, and l e t K^ and Kg be two non-empty closed subsets of pX - X. Then A p((X;Kg) = d ( X ; ^ , Kg) i f K^ C\ Kg=0 (1) ^(X;^) (2) o d l i K ^ A o d X j K g ) = o U X j ^ U K g ) i f K^fiKg^ 0 . 46 Proof :(1) I f K H K 1 = 0 , then by Lemma 1-2.1, we have 2 i eUX;^ ,K ) and *<(X;K )=s PUXJK^) Let 2 O<IX,K 1 ,K ). YX, * U ; K ) ^ YX. 2 By Lemma 1-2.1, J ( Y X ) and i s a subset of some set i n i s a subset of some set i n 2( YX). o((X;K , K ) * 1 YX; and 2 c^CXjK^ A ^(X;K ) = ^(X;K ,K ). 2 (2) I f E ^ H K 2 o^CX;^) ^ C 1 2 £ jZ), then by Lemma 1-2.1, oCU^UE^), o((X;K ) ^ * ( x j ^ U K 2 ) . 2 Let YX be any other compactification of X such that ot(X;X ) ^ TX and o^(X;K ) ^ YX. 1 Then 2 i s a subset of some set and K 2 fl K i s a subset of some set H i n •?{ YX) 2 i n -F{ VX). 3Fi, YX) i s a p a r t i t i o n of |3X - X, and Since and 2 YX be any compactification of X such that PUX;^) ^ So, 2 2 j£ 0, we have H ^(XjK^ K ) 2 1 = Hg. Thus K V KgSHj 1 YX. So rifXjK,,) A *(X;K ) = (X (X;K U K ) . 2 1 g 47 1-2.8 Lemma : Let X be l o c a l l y compact and l e t o<X be any compactif i c a t i o n of X. and that H , H 1 2 , Suppose , IL, J(<*X) g are non-empty closed subsets of respectively. Then the (3-family of o<X A o((X;K U K ) i s (J( o(X) - { H ,K ) )u { H U Hg} 1 2 1 2 1 Proof :By Lemma 1-2.4, there exists YX, a unique compactification of X such that 3( YX) = (J(oCX) - { H H D U f ^ U H }. r (a) o(X = 2 2 YX since every set i n J ( oCX) i s a subset of J ( YX). (b) ^ ( X ; ^ U Eg) ^ YX since K U K £ ^ U Hg 1 2 and so every set i n J ( ^(XjK^UK^)) i s a subset of J ( YX). (c) Suppose that SX i s a compactif i c a t i o n of X such e<X = SX and ^(XjK^UK^) = £X, then every set i n ^ ( o(X) i s a subset of some set i n 3 { SX) and every set of J ( o((X;K^ UKg)) ?( SX). i s a subset of a set i n Therefore H , H , 1 of some sets of J ( 8 X ) . R\C T e J ( SX) and H ^ U K c H 2 2 K 2 2 are subsets let £ U K c 5 €J(SX), j ? e J ( SX). Since K^c ^ , and 5( SX) i s a d i s j o i n t family of sets, we have T=U and so H ^ H ^ T € 7( SX). Thus YX = SX, and therefore 3( « A ^ ( X ^ U K g ) ) = J(<*X) - ^ H , 1 H^U^liH^. 48 1-2.9 Definition : A compactification dX of a l o c a l l y compact space X i s called a dual point of the l a t t i c e K ( X ) if dX ^ px and there exists no compactification YX different from both dX < YX 1-2.10 |3X s a t i s f y i n g dX and < pX. Theorem: Let X be l o c a l l y compact. Then dX. i s a dual point of K ( X ) i f and only i f there exist d i s t i n c t elements >p and q of )5X - X dX such that {p,q} ) = o^(X; Proof: (1) Necessity : I f dX i s a dual point of K ( X ) , dX <t px and there does not exist any T X ; then |3X ^ Since YX £ dX dX < {JX, dX < YX < px. such that . every set i n J{ (3X) i s a subset of 3(dX). Also, <*X £ px. (a) There exists one and only one set K i n 5 ( dX) that contains more than one element of 0X. ( i ) there exists at least one such set, otherwise dX = |3X. (ii) I f there exist two or more such sets namely: oKX;!^, ,.... ,K ,K ) n =o(X n and 2 , then /?X > ^ ( X ; ! ^ )>*<X 49 This contradicts the d e f i n i t i o n of <*X as a dual point of K(X). (XX = <X(X;K) So f o r some closed Kcr p x - X such that K contains more than one element, (b) cAX =o<(X;K), K = {p,q} where p,q are d i s t i n c t elements of |3X - X. Since K contains more than one element of /3 X - X, l e t K 3 {p,q}for some d i s t i n c t p,q e )3X - X . Suppose r i s d i s t i n c t from p,q such that r € |8X - X and r e K. Then ={p,q>is a set that s a t i s f i e s the following inequality: <XX = <rf(X;K) £ ^(X;^) £ /JX This contradicts the d e f i n i t i o n of <XX. (2) Sufficiency: p,q € pX~ X. exists Let oCX = <X(X;{p,q}) for d i s t i n c t Clearly, o(X < (3X. YX such that I f there dX < YX < j3X and ($X ^ YX ^ c<X then there exists F e J ( such that F c {p,q} Then F =(p) YX = px. YX) and F ?Mp,q} since P^X £ YX. or F = {q} and this implies that It i s a contradiction. 50 1-2.11 Theorem : Let tfX be a compactif ication of a locallycompact space X. Then J( *<X) has exactly one set which i s not a singleton i f and only i f (a) J3X and °<X ^ (b) there do not exist two d i s t i n c t compactifications TX and SX such that (/') both TX and o(X A YX = (ii) (iii) SX are dual points. c*X A SX ± <kX and the only dual point greater than TX A SX are TX and SX. Proof : (1) Necessity : Let J{ o(X) have exactly one set K which i s not a singleton, then (a) obviously, o(X ^ /3X (b) Suppose there exist two compactifications YX and (/) SX of X such that . both YX and SX are dual points, and (ii) dLX A YX = dX A SX ± o(X. Then by Lemma 1-2.10 there exist four points a, b, c, d i n /3X - X such that YX = <X(X; {a,b} ) and %X = Since << (X; {c,d} ). <*X A *<(X; {a,b} ) ^ o(X, a and b cannot both belong to K. Similarly, c and d cannot both belong to k. 51 There are e s s e n t i a l l y different cases t o consider : (1) a, b, c, d i K; (2) a, c (3) a, b, c 4 In the f i r s t and three t K b,d eK; K; d e l . case : <*X = oC(X;K) by lemma 1-2.7, o<X A o((X; {a,b} )= ', • <*(X;K, (a,b) ) and o<X A o<(X; {c,d}) = o( (X;K,{c,d>). By (if) above, o((X;K f {a,b} ) = so{a,b}= {c,d} which i s a c o n t r a - d i c t i o n since and o<(X;K, {c,d} ) YX = d{X; {a,b} ) $X = d ^X; {c,d} ) are distinct. In the second case : by Lemma 1-2.7, we have o(X A o((X;{a,b} ) = <*(X;KU{a) ), o(X A oC(X; {c,d} ) = o<(X;KU{c} ). By (ii) above, oC(X;KU{a}) = c<(X;KU{c} ) and hence K V {a) = K li {c) a ^ K , c So, b ^ d ^ K, because and s i n c e a=c. YX and SX are d i s t i n c t . Again, by Lemma 1-2.7, <rf(X; f a , b ) ) A <<(X; {c,d} )=tf(X; {a,b,d} ) . C l e a r l y , o((X; f b , d } ) i s a d u a l point greater than tf(X;{a,b,d}) and d i f f e r e n t f r o m both o<(X;{a,b} ) and <*(X; { c , d } ) . So (ii!) i s n o t s a t i s f i e d . I n the t h i r d case implies : Lemma 1-2.7 that <*X A o((X; {a,b} ) = c*(X;K,{a,b) ) and dX A e<(X; {c,d}) = Clearly <* (X; K U { c ) ) . o((X;K, {a,b} ) ^ ( X ; K U ( c l ) . Thus, c o n d i t i o n ( i i ) i s n o t s a t i s f i e d . So we c o n c l u d e that t h e r e do n o t e x i s t two c o m p a c t i fications YX and SX w h i c h s a t i s f y c o n d i t i o n ( / ) , ( i i ) and (iii). (2) S u f f i c i e n c y : From ( a ) , <<X £ e x i s t s a t l e a s t one s e t K i n pX, there J ( <*X) w h i c h c o n s i s t s o f more t h a n one e l e m e n t . ? ( o<.X) h a s a n o t h e r s u c h s e t H. Suppose Choose two d i s t i n c t p o i n t s a,b i n H and two d i s t i n c t p o i n t s e,d i n K. T h e n {a) = H, (b) { c ] c K, {dj c K. Let (i) YX = PC(X; {a,c>) and SX = <*(X; {b,d}) . H, 53 By Lemma 1-2.8, the 0 - f a m i l i e s o f o(X A YX and dX A SX are the same : C?(oCX) - f K,H}) U{K U H}. And a compacti- f i c a t i o n i s u n i q u e l y determined by i t s P - f a m i l y ; and c l e a r l y (ii) o(X A YX = o(X A XX o< X. By Lemma 1-2.10 both YX and SX are d u a l p o i n t s o f K(X) and by Lemma 1-2.7, TX A SX = d{X; {a,c}, {b,d} Let ). <X(X;{x,y} ) be a dual point which i s g r e a t e r than << (X; {a,c}, {b,d} ). Then e i t h e r {x,y}c{a,c} Equivalently, either o r {x,y}c{b,d}. o((X;{x,y} ) = YX or U{X; {x,y} ) = %X. Thus we have (m) the only d u a l points g r e a t e r than YX A SX are shown that YX and SX. Now we have YX and 5X are c o m p a c t i f i c a t i o n s of X s a t i s f y i n g (/), (/'/') and (/'/;) o f (b) above; which i s a c o n t r a d i c t i o n . So J(c(X) has e x a c t l y one s e t c o n s i s t i n g of more than one element. 54 1-2.12 Lemma : Let X be l o c a l l y compact, <*X a compactification of X with )5-family J( d X) and H a closed subset of /3X - X containing more than one point. Then eJ H i f and only i f ( * X ) <X(X;H) i <*X and there does not exist a compactification of the form d(X;K) such that <X(X;H) > <X(X;K) ^ eCX. Proof:(1) Necessity (a) : I f YLeS{dX) y P<(X;H) then ^ *X since every set of J ( o C ( X ; H ) ) i s either a singleton subset of /3X - X or H , so i t i s a subset of some set i n X X . (b) I f there exists a compactification of the form o((X;K) such that *<(X;H) > o<(X;K) ±UX then H f K and K £ T for some T e J( dX). Since J(<XX) i s a decomposition of ( 9 X - X , H 4 SidX). This i s a contra- diction. (2) Sufficiency : o<(X;H) ^ cXX implies that there exists T 6 J (d X) such that H<=T. Clearly i f H £ T, o((X;H) > <*(X;T) ^ (XX. It i s given that there does not exist a compactification of the form rt(X;H) so H > d(X;X) * dX. = T, . H € J( oCX). oC(X;K) such that 55 1-2.13 .Theorem : Suppose that X and Y are l o c a l l y compact and that T i s a l a t t i c e isomorphism from K ( X ) onto K(Y). Then there exists a homeomorphism h from j3X - X onto JBY - Y so that i f T{ <* X ) =o(Y then 5( <*Y) ={hIH] : H e J ( <<x)} . If p X - X consists of two elements, then there are two such homeomorphisms. I f card ( |3X - X ) £ 2, the homeomorphism i s unique. Proof :If /SX - X consists of one element, then i t is t r i v i a l . Suppose that px - X consists of two elements. Then K ( X ) consists of two elements : px and the one point compactification of X . flX, Hence K(Y) consists of two elements and i t follows that pY - Y consists of two elements. There i s only one isomorphism from K ( X ) onto K ( Y ) . are two homeomorphisms from However there px - X onto pY - Y, and "both s a t i s f y the condition of the theorem. Suppose that (1) px - X has three or more elements: Define a mapping h from px - X into pY - Y as follows : For a given point p e ^ X - X , choose any 56 points q and r i n /?X - X such that p,q,r are d i s t i n c t . By Lemma 1-2.10, we have d (X; {p,q} ) and d (X; |p,r}) are dual points of K(X), and thus P( o((X;fp,q} ) and r( o<(X; {p,r})) are dual points of K(Y). So By Lemma 1-2.1.0 again, there exist points a,b,c,din /3Y - Y such that T{ <*(X; |p,q})) = *(Y;{a,b> ) and r i * U ; {p,r})) = cC(Y; {c,d} ) By Lemma 1-2.7, o<(X;fp,q}) AX(X;{p,r} ) = o<(X; {p,q,r} ) so F( *(X; fp,q,r} )) = c<(Y;{a,b} ) A«<(Y;|c,d}) Now i f fa,b] fi{c,d} = 0 then by Lemma 1-2.7, <X(Y;{a,b}) A «(Y; {c,d} )= (Y; {a,b}, (c,d>) which i s a contradiction since there are three dual points of K(X) greater than o< (X; {p,q,r}) namely: o((X; {p,q} ) , o((X, {q,r} ), and o<(X; {p,r} ) while there are only two dual points of K(Y) greater than eUY; {a,b} ,{c,d} ) namely : c<(Y; {a,b} ) and c((Y; (c,d) ). Therefore {a,b}n{c,d} £ 0. But {a,b}?6 {c,d}because T i s one-to-one. Hence {a,b} D (c,d) i s a singleton. Let {a,b} A {c,d} =ia] . ¥e define h(p) = a. The element a does not depend on the choice of the elements q and r i f 0X - X contains more than three points. Suppose s e X - X and s i s different from p , q and r . fl Then there exist y and a i n p Y - Y such that 1 F(oC(X;{p,s) )) = ^ ( Y ; f y , z i ). We have T( and we may r U ( X ; {p,r} (X; {p,q} )) = d (Y; {a,bl ) assume that )) = oC(Y; {a,c} ). Using the argument given above, we have-fy,z} intersects both {a,b} Thus i f a 4 fy,z} and {a,c} i n one point. , then{y,z) = {b,c J. By Lemma 1-2.7, c<U;fp,q}) A o((X;{p,r) ) A <*(X;{p,s}) = c<(X; fp,q,r,s}) and oUYjfa.b} )Ao((Y;{a,c} ) A d (Y; *b,c) ) = oUY;{a,b,c} ). Therefore T{ d(X;fp,q,r,s})= (Y;Ja,b,c}) and there are six dual points of K(X) greater than <K(X; {p,q,r,s}) while there are only three dual points of K(Y) greater than (Y; {a,b,c}). This i s a contradiction. Thus ae{y,z}, so we have shown that f o r any S€(3X - X - ^ then a e {y,z}. if IUU;{p,s}))=*lY;fy,z>) 58 The mapping h defined above i s a homeomorphism: Let H be a closed subset of 0X - X which consists of more than one point. By Lemma 1-2.11, ( i ) r(e((XjH)) =oUY;K) for some closed subset K of j3Y - Y which contains more than one point. Let p and q be any two d i s t i n c t points of H, and T( <X (X; {p,q} )) = *(Y; {a,b} ). by Lemma 1-2.2, And then that i s Then <x(X;(p,q}) ± U{X;R). f ( *(X; fp,q})) = f(<X(X;H)) o((Y;{a,b}) asc((Y;K), hence (a,b}=K. But hC {p,q}3'. s {a,b} . Thus (ii) h(H) = K. Now define a function k mapping into 0Y - Y pX - X i n the same way as h : Given a e pY - Y, choose d i s t i n c t b,c e |8Y - Y. This i s possible as K(X), K(Y) are isomorphic and p x - X contains more than two elements. By Lemma 2-1.10, o((Y; fa,b} ) and <X(Y; {a,ci ) are dual points of K(Y) and thus r ~ (o((Y; {a,b})) 1 and r " ( o( ( Y ; {a,c})) 1 are dual points of K(X). Therefore there exist points p,q,r,s i n /SX - X such that IT (o((Y; {a,b})) = o((X;(p,q}) 1 59 and r " (eA ( Y ; {a,ci)) = d ( X ; {r,s>). 1 {p,q}fl{r,s} consists of one point, by the same argument as above. Let fp,q}n?r,s} = ir] . The point r does not depend upon the choice of b and c. Define k(a) = r . By the same argument as f o r h, we have ( m ) k ( K ) c H for the sets H and K of ( i ) . Now l e t p,q be d i s t i n c t points i n px - X , then 37( oi ( X ; {p,q})) = o((Y;{a,b)) f o r some d i s t i n c t a,b i n P Y - Y . We may assume h(p) = a. Suppose k(a) ^ p, then by (/if) k(a) = q. • Choose a point r e px - X d i s t i n c t from p and q, then there exists c«pY - Y such that r ( * ( X ; {p,r})) = o U Y ; fa,c}). Clearly k(a)€ (p,r}.But we have k(a) = q which is distinct from p and r . So i t i s a contradiction. Thus k(a) = p and k-h i s the identity mapping of p X - X . Similarly h'k i s the identity mapping of PY - Y ; and k = h" 1 . Now by (ii) and {iii), we have (iv) i f r ( * ( X ; H ) ) =o((Y;K), then hCH] = K. Now h i s a closed mapping, and similary so is h . Therefore h i s a homeomorphism. 60 TX i s a compactification of X with Suppose 3( Y X ) , and T( Y X ) = TY. J5-family Then ? ( YY) = fh.LHJ : H € 3( Y X ) } : Let H 6 J ( TX) and H contain,, more than one Then oC(X;H) = Y X . By (iv) above, point. c<(Y;h(H))^ YY. By Lemma 1-2.12, there exists no compactification of the form <*(X;V) o((X;H) > such that *C(X;V) * TX. So there exists no compactification of the form o<(Y;W) such that ctf(Y;h(H)) > o<*(Y;W) ^ YY. By Lemma 1-2.12 again, h(H) e J ( YY). i f K e J(YY) Similarly, and K consists of more than one then h" (K) £ J ( Y X ) . element, 1 Now consider a singleton /p} i n J ( Y X ) i f I f fh(p)} i s not i n 3( Y Y ) since i t exists. h(p) e 0Y - Y and 3 (YY) i s a decomposition of PY- Y, there exists a subset K containing more than one point i n 3 (Y Y) such that h(p)e K. —1 Then p £h —1 UL1 where h than one element. since h" CK3 e 1 IK1 contains more This i s a contradiction J(YX), and the sets i n 5"(YX), are mutually d i s j o i n t . Similary, if{a) e J(YY), then { h ~ ( a ) h 3 { Y X ) 1 and so ^(YY) = {hCHI : H e J(YX)}. 61 (4) The homeomorphism h i s unique: Let t be a one-to-one mapping of j3X - X onto that f o r any compactification r(TX) = YY, then J(YY) = j8Y - Y such Y X of X , i f e ft£H):H Given J(TX)}. p e px - X and choose two other points q, and r i n px - X such that p,q,r are d i s t i n c t . Then there exist d i s t i n c t a,b,c i n (3Y - Y such that r( o<(X; {p,q})) = c*(Y;{a,b} ) and T( o<(X;{p,r})) = o<(Y;{a,c} ). We have h(p)=a, and At the same b ^ c as T i s one-to-one. time, t l f p , q } l = {a,b}. So t(p) = a or t(p) = b. Suppose t(p) £ a, then t(p) = b. But tfp,rJ = la,cj. and we had t(p) = b £ |a,c}. So i t i s a contradiction. Therefore t(p) = a = h(p) or t = h. This completes the proof of this theorem. 1-2.14 Theorem : Suppose that X and Y are l o c a l l y compact and that h i s a homeomorphism from j8X - X onto j8Y - Y. Let dX be a compactification of X with j?-family 3(oiX). ion Then there exists a unique compactificat- o<Y of Y whose /3-family i s {hOD : E e J{ P<X)}, and the mapping 7 defined by I( i/,X) =«<Y i s a l a t t i c e isomorphism from K ( X ) onto K(Y). Proof :Let f^ be the p-function mapping ^ X onto o<X. 62 Then f^-h"^ i s a continuous f u n c t i o n mapping / J Y - onto o<X - X. Let Y <KY = Y U [ o(X - XI, and d e f i n e a —1 f u n c t i o n k mapping JJY onto f o r p e f3Y - Y °(Y by k(p) = f^ • h~ (p) :, k(p) = p f o r p i n Y . Let o(Y have the quotient topology induced by k. (1) o(Y i s compact w i t h Y a dense subspace:i<Y has has the quotient topology induced by k, so k i s continuous from j3Y onto tfY. So «<Y i s For any open set Og i n <*Y, there compact. e x i s t s an open set 0^ i n klO^JCOg. 0Y such that Since Y i s dense i n ^Y,there e x i s t s p e Y 0 0^ such that k(p) = p € 0^ . So Y i s dense i n o<Y. oCY since Y i s open dense an open subspace o f in I t i s c l e a r that Y i s f$Y and o<Y has the quotient topology induced by k. (2) o<Yis Hausdorff: For d i s t i n c t p o i n t s p and q i n o(Y we have three d i f f e r e n t (/*) cases: .p 6 Y and q 6 o<Y - Y - o(X - X: Since Y i s l o c a l l y compact, Y i s open in j J Y , k ~ ( q ) i s c l o s e d i n /SX and, 1 so i s compact. There e x i s t open subsets U and V o f j3X such that p e TJ c Y , k" (q) e V, and U A V = 0 1 Let U*= kCU] , V* = k m . 63 It follows that U and 7 are d i s j o i n t open subsets of d~L containing p and q, respectively. (ii) Both p and q belong to Y : Y i s open i n |3Y, there exist open subsets UcY, V c l f j3Y such that 0 UD V = 0 and peU, q e V. Since Y i s open i n c<Y and U = k~ CUT, V = k" .CVl 1 1 U and V are open i n o<Y. (iii) Both p and q belong to oCY - Y : <*X - X i s closed, so i t i s compact and regular; therefore there exist two open subsets G and G of f*X - X q. P containing p and q respectively such that C1G f l C1G = 0. Then k" (ClG ) 1 and k~^(C1G ) are d i s j o i n t closed and hence compact subsets of 0Y - Y. Then k" (ClG ) and k~ (ClG: ) are 1 1 d i s j o i n t closed subsets of /3Y which i s compact, normal. There exist d i s j o i n t open subsets H and H of * p q J fJY such that k ~ ( C l G ) £ H and 1 p k" (ClG ) S H . 1 Pi. _ k PJ. p Since k~ (G ) and 1 P A (G ) are open subsets of j3Y - Y, there exist two open subsets U and 64 U of (3Y such that k~ (G ) = U n (jSY-Y), P P * ' (G ) = U fi ( BY - Y). 1 q k~ r 1 q q K Then we have k ~ (G ) = H f) U n ( B Y - Y) P P P k~ (G ) = H ft U n (pY - Y). 1 K 1 Now l e t , V = (H A U 0 Y) U G and P P P = (H H U 0 Y) U G . Then V , V V q q P q q p q are d i s j o i n t subsets of e<Y, containing p and q respectively. k" (V ) = k" (H 0 U A Y ) U k" (G ) 1 1 1 p = p p p ( H n u n Y ) U (H n u n ( R Y - Y ) ) p p = (H n u ) p p p p H p IY U (BY - Y)] V K p which i s an open subset of (5Y. oCY has the quotient topology by k; V induced i s open i n <*Y. S i m i l a r l y , i s open i n e<Y. (iJi) we have ion of Since Prom (/), (;j) and o<Y a Hausdoff compactificat- Y. 3 ( O C T ) ={htH3 : H e 3F( dX)} : Now k i s a continuous function from |3Y onto oCY which leaves the points of Y fixed, so k must be the p-function of <*Y. Let K be a set i n the p-family J(o<Y) of «<Y. Then there exists a point p i n o C Y - Y = o ( X - X such that K = k" (p) = l f ^ - h " ) " C p ) = h ( f * ( p ) ) . 1 1 1 1 Thus K i s the image under h of a set i n J(<*X), 65 the p-family of (*X. On the other hand, i f H i s any set i n ^ ( r f X ) , there exists a point q in o<X - X such that H = f ^ ~ ( q ) . 1 Then h(H) = h ( f , ~ ( q ) ) = U r f - h ' W ) = k' (q) i n 1 2(cXY). 1 Thus d Y) = { h(H) : H e 2 U X)} . The uniqueness of <XY follows from the fact that a compactification i s uniquely determined by i t s p-family. (4) Define T{ d X) = <XY, then T i s a l a t t i c e isomorphism from K(X) onto K(Y): If YX and SX are two different compacti- f i c a t i o n s of X, then the two J ( YX) and p-families .?( SX) are d i f f e r e n t . Consequently, {hCHl : H 6 J ( YX)} £ { hQO : H € J ( §X)} which implies that T( YX) ^ T(SX). Thus F i s one-to- one . By the same argument, i f YY i s any compactification of Y with p-family 3-( YY), then there exists a unique compactification of X such that YX J ( Y X) ={h" (H): H e J ( Y X).} 1 Thus r( YX) = YY, and i t follows that T i s a bisection from K(X) onto K(Y). The fact that P i s a l a t t i c e isomorphism i s a consequence of the fact that the following statements are successively equivalent : YX ^ SX; each set i n J(SX) i s a subset of a set i n J ( YX); each set i n {h(H) : H e J ( % X)} bb i s a subset of a set i n {h(H) : H € 3= ( YX)} ; T( YX) ^ T( S X). Thus the proof of the theorem i s complete. 1-2.15 Corollary : Suppose that X and Y are l o c a l l y compact. Then the l a t t i c e s K(X) and K(Y) are isomorphic i f and only i f |8X -X and ]3Y - Y are homeomorphic. CHAPTER I I HOMEOMORPHISMS OF p N In this chapter, we study homeomorphisms onto (3N. 2-1.1 Definition: If fl^ , &2 a r - u l t r a f i l t e r s on countable sets X e and Y respectively, then Q.^ and 0,-, a r e s a i d to be of the same type i f there i s a one-to-one correspondence f of X onto Y such that f o r each Ec:X, Ee i f and only i f fCE] e Q ; or equivalently, 2 pe |3X, qe (3Y are of the same type i f E c X , p e C l ^ E i f and only i f q C I D V f CE]. pi 2-1.2 Definition: If K i s a permutation of N, fl^ i s an u l t r a f i l t e r on N, fig = K( the sets ) i s the u l t r a f i l t e r which contains Jl(E) f o r a l l E eff^. 2-1.3 Remark: It follows that i f two u l t r a f i l t e r s D.^ , ft on N 2 are of the same type, then there i s a permutation K of N such that n 2 = 71(1^). There are 2 C types of u l t r a f i l t e r s on N, each type contains c u l t r a f i l t e r s ; since N has 2 u l t r a f l i t e r s and c permutations, and no type can contain more than c u l t r a f i l t e r s . 69 2-1.4 Theorem: Every homeomorphism of pN induces a permutation of N and every permutation of N induces a homeomorphism of pN. Proof:( 1 ) Since the points of N are the only isolated points of (3N, and isolated points must be mapped to isolated points by a homeomorphism; so every homeomorphism of pN induces a permutation of N. ( 2 ) I f K i s a permutation of N, then it i s a continuous function from N into pN. Thus ft , the extension of K to pN i s a continuous function from pN into pN (By Theorem 0 - 2 . 4 ( 1 ) ) . K* i onto since ix t (JN.1; i s compact and N i s dense i n pN. s For d i s t i n c t x,y i n pN, there exist d i s j o i n t A, B subsets of N such that because x € Ol^jjA , y e Ol^B pN i s Hausdorff and i ^ l ^ A:AcN) forms a base f o r pN. (Remark 0 - 3 . 2 ) 7C (x) € C l p * CAI and 7t*(y) £ C l p 31 CBI . N Since * EA3 Hit LB] = 0 sets of N, so Thus and n l A l , n CB] 0 1 1 T £A1 r i C l N B N ft.IB] = 0. 7t (x), K (y) are d i s t i n c t . i s a homeomorphism of are zero- Therefore 7T pN, and i t i s the only homeomorphism of pN which coincides with 7t on R. It follows tnat there i s a natural isomorphism between the group of a l l homeomorphisms of pN and the group 70 of a l l permutations of N . So 0N has p r e c i s e l y c homeomorphisms. 2-1.5 Theorem: Por f r e e u l t r a f i l t e r s , on N, there i s a homeomorphism of (3N which c a r r i e s only i f fl^ and a r "to O.^ i f and isomorphic as p a r t i a l l y e ordered s e t s . Proof:As a r e s u l t of Theorem 2-1.4, we need to prove that = Jt ( there i s a permutation n of N such that ) i f and only i f Q.^ and Cl^ are isomorphic as p a r t i a l l y ordered s e t s . (1) N e c e s s i t y i s obvious. (2) Sufficiency: If and a r isomorphic as e p a r t i a l l y ordered s e t s , there i s a mapping f of onto only i f f'EAD c f l B l , let H n = N - {n} one-to-one SXp such that A c B and c l e a r l y i f and fCNI = N. , c l e a r l y the H are n maximal proper subsets of N and are permuted by f . Now d e f i n e a permutation TT (n) = m n o f N such that ±'(H ) = H . For any E « i , n m• i F = N - E, then E = H H . Let ' neF n E if ' = n?F f ( H ) . n F(E)cf(H ) n Since E c H n for a l lneF; f o p ^ n e F > so F ( E ) c E ' . let 71 for a l l neF, S i m i l a r l y , E'c:f(H^) f " (E' ) c H f o r a l l neF; 1 n and t h e r e f o r e so f" (E')cE 1 f ( E ) = E*. n = n (B- „<„)) - N -y^„) But >• - % n f(H ) = n = FH„ n ( n ) ? n N - {n ( n ) : n e F} N - f(E) = ( ( n ) : n e F} , so JTI N - E* = F that i s I N - E ] so K f(E) = 71 (E). The theorem f o l l o w s . 2-1.6 Theorem : If £ are u l t r a f i l t e r s on N, then there i s a permutation TI o f N such that X ( I i ) = £i 1 and 1 71 ( _p_) ^ 2 il 2 Proof :il^ ^ complement Hg, there e x i s t s E^c N such that E 1 £ ^ and with i n f i n i t e N - E^ € (By Theorem 0-1.6) Choose £Lj £ i l such that N 2 i l a r l y there e x i s t i n f i n i t e Eg e Ug, E^ e Si^, that e 12^. Sim- sets Eg and E^ such EgU E^ = N - E ,and t E n E^ = 0. 2 Let 7[ be a permutation of N such that X (n) = n for n e E^, and 7t(n)eEg i f neE^. and u ( lig) ^ li-g. n(n) Clearly i f n e Eg n( si^) = 72 2-1.7 Corollary : If a, b are d i s t i n c t elements of (3N then there exists a homeomorphism that 2-1.8 7f (a) = a and 7T of (3N such K (b) £ b. Theorem : If A' , B' are non-empty open-closed proper subsets of N , then there i s a homeomorphism of £N which maps A' onto B*. Proof :A', B 1 are non-empty, there are i n f i n i t e A, B * subsets of N such that c l ^ A fiN = A' , c l ^ B f i N * = B' A' ^ N* £ B are f implies that N* - A' and N* - B' non-empty open-closed subsets of N . Thus N - A, N - B are i n f i n i t e subsets of N. Let and (Theorem 0-4.1) 7i be a permutation of N which maps A onto B 71 induuces a homeomorphism of /3N. 73 CHAPTER I I I HOMEOMORPHISMS OF N In t h i s chapter we study the homeomorphisms o f * N . Most o f the r e s u l t s o f t h i s chapter depend on those of chapter I I . The continuum hypothesis i s assumed out t h i s chapter. through 74 3-1.1 Theorem : N* has 2,c of N P-points, and the set of P-points i s dense i n N . Proof (1) There exists P-points i n N . Let {WJ} , the family of a l l the open- closed subsets of N Wj = N , he well- ordered with where <*. runs through the countable ordinals. Now subsets of N A» {A^} select a family of open-closed i n the following way : = N For o< , a countable ordinal such that an open-closed set A£ has been selected for p < <>< so that each fl { A£ : i s not empty, by theorem 0-4.2 a non-empty open-closed set 0 < * } = B* there exists G^' such that Ci . B. Q* C 1 If C; n 0, set AJ = If CJ 0 WJ £ 0, set A,J = G' 0 WJ = The family ordinal} { AJ 1 : runs through the countable so formed i s l i n e a r l y ordered by set inclusion. Let A* = H |A]< : c< runs through the countable ordinal} (a) A* £ 0 since A ^ are closed subsets 75 * o f t h e compact s p a c e N (b) with the f i n i t e i n t e r s e c t i o n property. A' = ( a } . I f a £ b» € A', t h e n 1 f a' 6 ¥' , b* e ¥' f o r some r , p o f r P the countable o r d i n a l s such t h a t * ¥' A « * = 0 s i n c e N i s H a u s d o r f f and |WJ} f o r m s a b a s i s o f N ; and t h e o f AJ shows t h a t i f A'H ¥ | £ choice 0, t h e n A » c A j c ¥ ^ , and i f A»H ¥J ^ 0 , t h e n A'c: Al,cr¥'. T h i s i s a c o n t r a d i c t P P ion. claim that a (a) 1 i sa P-point: The f a m i l y { A j } where ai r u n s t h r o u g h the at a countable 1 o r d i n a l s , forms a b a s i s s i n c e f o r any open n e i g h b o r h o o d CT o f a ' , t h e r e exists a countable o r d i n a l K s u c h t h a t a'e W^CG, a (b) 1 so 6 AJcr¥J c 6 . If [G ± : ieN} i s a countable collect- i o n o f open s e t s c o n t a i n i n g a ' , t h e r e exists the <*± s u c h t h a t A ' . c e . , I f <* i s s m a l l e s t o r d i n a l which exceeds e v e r y o t i , i e l f t h e n A^ cr ^ ^ i * * So a* i s a P - p o i n t of N . 76 There are 2 c * P-points i n N . T h i s i s shown by the f a c t that i n the preceding c o n s t r u c t i o n , there are at l e a s t disjoint candidates f o r each c stages. 2 i n each of the In Theorem 0-4.2, the c o n s t r u c t i o n of A shows that we can have B with the same property as A, and m^ e A^ f\ A n k 1 + € A i n B f i A = 0 by choosing k 0 .... f\A - j U n^ and 2 i k A 2 n " - - ° k A n A 1 k-M " h^1 ( t h i s i s p o s s i b l e because A^ fl A is infinite). 2 "h 0 ... 0 A^. H A ^ ^ Then A, the set of a l l n^ and B, the set of a l l m^ chosen chosen are d i s j o i n t sets with the r e q u i r e d property. The set of P-points of N i s dense i n N T h i s f o l l o w s from Theorem 2-1.8 that f o r any non-empty A', B* proper open-closed * of N , there i s a homeomorphism of subsets which maps A' onto B* and a P-point i s mapped to a P-point. 77 3-1.2 Theorem : N contains non P-points, and N i s not homogeneous Proof :If every point of N i s a P-point, then the inter- section of any countable family of open sets i s open; so the union of any countable family of closed sets i s closed. Then every countable subset of N i s closed and discrete ( by Theorem 0-1.16(2) ). This * i s impossible because N ±s i n f i n i t e compact. N i s not homogeneous since no homeomorphism of *, # N can carry a P-point to a non-P-point of N 3-1.3 Theorem : If a', b 1 are P-points of N homeomorphism of N f then there i s a which carries a' to B'. Proof :Let {AJ}, with o< running through the countable ordinals be a well-ordered family of a l l the openclosed subsets of N that contain Aj = N* a' such that and l e t Similarly, { B'} i s f o r b , 1 B' = N* V = and - BJ . Now construct a permutation ip of the family of a l l 78 open-closed subsets of N (a) lp (A' )c<p ( B ) (b) (pmaps {A^} Let 9(Aj) = Bj such that i f and f onto only i f A' c B » [Bj] . <P(X') = Yj , and proceed by trans- f i n i t e i n d u c t i o n as f o l l o w s : suppose <P (AJ< ) c< i s the s m a l l e s t o r d i n a l f o r which has not yet been defined such that the sets f o r which 4> has been d e f i n e d form an at most countable (R.| , and ring ip preserves f i n i t e unions, finite i n t e r s e c t i o n s , complementations, and i n c l u s i o n s . Since a', b' are P-points, and BJ ( f > <*) such that i n t e r s e c t i o n of a l l A£ and BJ <P (AJ there are sets AJ AJ c A ^ w i t h A} f o r which ip has been d e f i n e d , i n the i n t e r s e c t i o n of the corresponding ) . if{A\ Define ) = BJ <f(XJ. ) = YJ the s m a l l e s t r i n g XJ c o n s i s t s of a l l sets of the form and f i n i t e unions of tnese s e t s . or RflAJ = -0 ; so we and : R«^l 1 0 X' . )UR 2 ROAJ ) = ip(R) I?(RUAJ ) = ip(R)UBJ n , R : Y; ( p i R ^ X J U R g ) = lf> ( R ^ O Y J and , ROXJ and ^<P(R ) . 2 (Red^) RflAJ = AJ. cp on Let q>U nxj (R^ , AJ Either need to define : R€(R.1 } , {(R { AJ UR that contains <R2 sets and Now {R n x ; i n the 2 € } 79 The f u n c t i o n (R-2 a n ( if i s now d e f i n e d on a countable r i n g i y preserves f i n i t e unions, f i n i t e inter- s e c t i o n s , complementations and i n c l u s i o n s . Now, d i v i d e the members o f (ft i n t o three c l a s s e s 2 {F-}, ( G J , {E^} such that A^F,, i n c l u s i o n holds between Put S and = F^PgO ± j C A * a n d n 0 f o r a l l ke N. HF T . = G U G„ V G i UG. then we have and H k* j' T f o r a 1 k,jeN 1 H c T. then H c A^. k Similarly, k tf(T^ , forT c A V(T )f H k fc and i f fora l l i.keN; p y ( S ) f ... ^ ( S ^ and n n 9(1^)^^(1 ) f o r a l l i , 3 , k € N. WS^VU^) By Theorems 0-4.4, 0-4.5 there e x i s t s an open-closed set Z inN d i f f e r e n t from any s e t so f a r i n the range o f f such that (f(T.)cZC^(S.) n e i t h e r o f the sets and (f(Q)DZ £ 0 f o r a l l i , j , € N, <f ( H ) and Z c o n t a i n s the other k whenever QHA* ^ 0. Since A <= A*, B , c Z and Z i s a member o f the f a m i l y {B„} . (fUj Let = Z and ip(X„) Y Define = N* - Z . (R^ be the r i n g generated by |R, and A^ : 2 c o n s i s t s o f a l l sets o f the form ROA* , Rn X A (Re <ft_) and f i n i t e unions o f these s e t s . Define q>(RO = </>(R) n cP(A*) , flRHXj = <P(R)0 <p(PiVQ) = ¥(P)U<P(Q) i f <P(P) and are d e f i n e d . dl^ In t h i s way «?(X^) , and tf> i s extended to the r i n g so as to preserve f i n i t e unions, f i n i t e s e c t i o n s , complementations and C l e a r l y a'e R Re (R^ . cf~ inter- inclusions. i f and only i f V e <f(R) Similarly cp(Q) for a l l i s defined to s a t i s f y the 1 same i n d u c t i o n hypothesis f o r the f i r s t member of { } which i s not yet i n the range of tp and i s extended to a r i n g as above. Since if i s constructed such that <p (A' )<z f ( B ) 1 * i f and only i f A ' c B 1 and only one h(a) € N for (a) , f o r any a e N there i s one which i s contained i n ¥>(A') every open-closed set A ' which contains a : if i s d e f i n e d on every open-closed set of N :- Suppose X i s an open-closed subset of N , c l e a r l y N - X i s a l s o open-closed i n N or a ' f N - X. . Hence so i s <p~^ . So <f i s ; e i t h e r a'e X d e f i n e d on X; (/> permutates and the f a m i l y of a l l # open-closed sets of N . (b) For any a e N , let iA ) rf open-closed subsets of N c l a i m that be the f a m i l y of a l l that contains a. P\ {k^) = {a\ : We Por any a^b e N , there e x i s t open-closed A, B such that a e A, b e B = 0 ; that means and A.HB there e x i s t s A e {A*} such that b b (c) f){Aj . 4 f] jlfCA^)} i s non-empty. of A; so {<f(A^)\ open-closed subsets of N i s a family which i s compact. Since there e x i s t s B, an open-closed s e t such that B e f a m i l y of n (A^.} {A^} f o r {A^.} Thus fll^tA^)} (d) Let b fl{cf ( A j } 6 sub- flj^U^)} that ; so <f(B)c {lf(A^)) has the f i n i t e is , any f i n i t e i n t e r s e c t i o n property. i s non-empty. , then ft {<f (A^)} = {b) . any b p c € N , there i s an open-closed C For such that c € C and b 4 C. Thus C 4 {^(A^)} s i n c e b € HfyU^)}. Now (f" (C) and if" (N* - C) 1 1 are complementary. So e i t h e r if~ (C) o r 1 vf " (N* - C) belongs t o (A^)and 1 not, therefore i f " (N - C) £{A*}. i f " (C) does 1 There i s D = l f " ( N * - C)e { A j s u c h that c 4 v(D) 1 So c 4 nfvuj} (e) Define h(a) = b . Since If maps |B^ } {A^} onto , h(a*) = b' and i t i s c l e a r that h so d e f i n e d i s a homeomorphism. proved. The theorem i s 82 3-1.4 Theorem : * c N has 2 homeomorphisms. Proof :U ) There are 2° p-points ( Theorem 3-1.1), bytheorem 3-1.3, there exist at least 2 C homeo- morphisms of N . (2) Every homeomorphism of N induces a permutation # of the c open-closed subsets of N and d i s t i n c t homeomorphisms induce distinct: permutations. * N 3-1.5 c has at most 2 homeomorphisms. lemma : If X i s a countable discrete subset of N then f o r each a e X there exists E' a a e E' and { E 1 , so that : a e X } i s a family of d i s j o i n t sets where E c u and { E : a e X } i s also a family of d i s j o i n t sets. Proof :Since X i s a countable subset of N , l e t X = | a,j, &2 > ••••}• Now l e t us construct E^ as follows : (a) E 1 : {a^} i s open i n X, so there i s a 1 € F' c and Fj f) X = { a ^ . Set E 1 N such that = F . 1 83 (b) E : 2 | a } i s open i n X, so there i s ~F^a N such that 2 a 2 eF and 2 that a fGl 2 2 a a 2 € E p N V pN ( = F F E (c) E ^ while E 0 ( N 2 ^ ^ 1 £ 0 1 ( _ ^ ) , N A 2 n " 1 E N and J <i , 1 i s d e f i n e d whenever j< i : = F i The n ( N - d u 1 V) 1 .) E and i ^ - jWl J S e a. * 01 ^ F.. t E.) . f a m i l y { E ^ : i € N } constructed conditions 3-1.6 P iii lE such that Since a - 4 E'. whenever F!flX = f a.}. Thus, a. € 01 „ N (F. fl (N 1 E. D - E ). a. € C l ^ (N - . U R N Theorem 0-2.4(4). ) { a^ } i s open i n X, so there i s F ^ c N a. € F! implies 2 o r Set 2 Since a £ 2 V C C 1 e F£fiX = { a | . s a t i s f i e s the required. Lemma : I f XUY i s a countable d i s c r e t e subset of N and a' i s i n the c l o s u r e of both X and T, then a' i s i n the c l o s u r e o f XOY. Proof :Suppose on the c o n t r a r y that there i s an a'e E , and E ' f ! (XflY) = 0 so that 1 E' = CI N E«=K where EHN*. X U Y i s countable and d i s c r e t e , so by Lemma 3-1.5, 84 f o r each b e X UY and there e x i s t s EJ such that b {E£ : b € X UY} E^C N where and i s a f a m i l y of d i s j o i n t {E fe d i s j o i n t sets also. E sets : b e X U Y ) i s a f a m i l y of Let = {n e N : n e ( E n E) f o r some b e ( X f l E )} 1 x b = {n;e N : n e ( E 0 E) f o r some b e (Y HE')} . E Since a b e EJ b b i s a l i m i t point o f X and a'e E , a' i s a 1 1 l i m i t point o f X HE' (that i s a' i s i n the c l o s u r e of X / 1 E ' ) . E' i s an open-closed s e t c o n t a i n i n g -X. XHE . 1 Hence, Cl a w ( X D E•)D N * C E* and so a'eE'. X X pi* Similarly, a ' e E . But E A E = 0 : y' x y p e E H E , then by d e f i n i t i o n , y suppose 1 ; x pe(E HE) f o r some b e (XH E') p e (E n E) f o r some a e (Y H E ) . b 1 p e (E n E. ) . E ' H E ' =0 x y 3-1.7 ; and So, But E , E,are d i s j o i n t . So that i s a contradiction. Theorem : I f X i s a countable i n f i n i t e d i s c r e t e subset o f N* f then Clp N X = p X , C1^ X i s homeomorphic to p N . N Proof :Let 71 be any one-to-one correspondence from N onto X . 7C i s a continuous f u n c t i o n from N i n t o of 7T to p N i s Clpjj X. (a) continuous from p N i n t o 01^ (b) from p N onto C l ^ j j X, since (c) one-to-one Then 7[ the extension : since X; 7t ipN3 N i s Hausdorff. i s closed; I f x, y are K : d i s t i n c t elements o f GN such that (x) = TT (y) , then f o r any neighborhood 7T*(x) i n C 1 ^ of N U X, there are N ( x ) , N(y') neighborhoods o f x and y i n p N such that N(x)0 N(y) = JT*£N(x)]cu 0 , and 7[*[N(y)l C U . Then we have 7t*(x) € 0 1 ^ 7C*[N(x) n N3 /(y) e CI N * * I N ( y ) AN] . 7T CN(x)ON] U3i 'CN(y) Now, and ftND i s a countable * d i s c r e t e subset o f N ; C 1 pN Cl JI*IN(x)AN3 = C1 N # 7T*CN(y) AN] = C 1 ^ p N N 7T*£N(x)nN] and 7i*CN(y)nN3. By Lemma 3-1.6, -K*(x) € CI N {it* CN(x) fi NI H 7C* CN(y) AN]} i s empty s i n c e 7C IN i s one-to-one N(x) A N(y) = (d) So which and 0. closed. G lp^^ Since i s homeomorphic to pN. ji i s a homeomorphism of N onto X, both of N and X are d i s c r e t e countable, then 7 1 ' , the e x t e n s i o n o f 71 from pN to |JX i s a homeomorphism. So G l p j j X i s homeomorphic to |3X; o r e q u i v a l e n t l y , CXpjjj. X = p x (by the uniqueness of p x f o r given X). 86 3-1.8 Theorem : I f f o r each n c N , h onto N and (IT } and n i s a homeomorphism {^(tM)} a r e of N countable •it- f a m i l i e s o f d i s j o i n t open-closed subsets of N , then there i s a homeomorphism h of N t h a t f o r each n e N and onto N such x e U ' , h(x) = h ( x ) . n' n v Proof :Let x^ e Uj y € h (U|). 1 1 We d e f i n e be a P-point, y 1 = h (x^) ; 1 Let fi.^ be the r i n g generated by IT U {tP} ip a mapping of ft^ i n t o the f a m i l y o f a l l open-closed s e t s of N i n the f o l l o w i n g way: for ip(U£) = h ^ U p a l li €N , 9(0) = 0 , V(N*) = N*» if(N - ip(P) whenever cp(P) i s d e f i n e d , - P) = N f ( p UQ) = ¥ ( P ) u <P(Q) f ( P A Q ) = <p(P)0(p(Q) » whenever «p(P) and i p ( Q ) are defined. Now ip i s d e f i n e d on (R^ such that f i n i t e unions, f i n i t e i n t e r s e c t i o n s , and i n c l u s i o n s . Let {W^}. be c l o s e d sets c o n t a i n i n g x^ , {uj} - {ip(R) : r e s p e c t i v e l y , where fie^] complementations, the f a m i l y of a l l openiU^} be the f a m i l y o f a l l open-closed s e t s c o n t a i n i n g y^ . and (/> preserves Arrange i n order as (WJ} - ( R j {A;} and <?< runs through the countable {B^} 87 ordinals. We extend (f to the ring generated by {A j}u(R onto the ring generated by ( 1 {BJ}u{ip(R) Suppose : Re R^} i n the following way : <<^ i s the smallest ordinal f o r which <P(AJ[) has not yet been defined such that the sets f o r which if has been defined form an at most countable r i n g ^ and if> preserves f i n i t e unions, f i n i t e intersections, complementations Define Q = AJ^ f\ U n if>CU - Q ] = h CU n n n n n n inclusions. f o r each positive integer n; - Q 3 , and extend ip to the ring N <ft generated by -CU - Q > U -CQ^ U R « 5 and n 2 Then extend <P to the ring generated by {AJ^} u (R^ by the same method i n Theorem 3-1.3. Then the theorem follows. 88 3-1.9 Theorem : Suppose X and Y are countable s e t s o f P-points of N and p and q are l i m i t respectively. N p o i n t s o f X and Y There i s a homeomorphism o f N onto which c a r r i e s p t o q i f and only i f p and q are o f the same type. ( D e f i n i t i o n 2-1.1) Proof :(1) N e c e s s i t y : X and Y are d i s c r e t e (Theorem •jt- 0-1.16(2)). onto N Let h be the homeomorphism of N such that h(p) = q . S e l e c t subsets X^ and Y^ o f X and Y r e s p e c t i v e l y so that X - X^, infinite Y and p € C l ^ X ^ Y - Y^, are , qe 0 1 ^ Y^. L e t N = h C X ^ H Y 1 and X = h " CY^ . By 1 2 g Lemma 3-1.5, q e C 1 ^ 01 p N N Y^ and h l X ^ ^ h C X ^ ^-Cql, then qe 01 ^ Y and p c 0 1 ^ ^ X 2 2 , . C l e a r l y h induces a one-to-one between X ? and Y 2 correspondence such that f o r each E<=X , 2 and p C l ^ E i m p l i e s that h ( E ) c Y , so e 2 q eClp N h(E) . Let q be any one-to-one pondence from X -X the one-to-one 2 onto Y - 1^. Define f as correspondence from X onto Y such that f ( x ) = h(x) f o r x e X f(x) corres- = g(x) f o r x e X - X 2 and 2 I f E<=X, then p e C l ^ E i f and only i f 89 p i s a l i m i t point of E D X^ ; hence p e C l ^ E i f and only i f q i s a l i m i t point of f(E) fiY^. Therefore p and q are of the same type since i f EcX, p € Clp Sufficiency : E i f and only i f y e 01^ N f (E). Let h be a one-to-one corres- pondence between X and Y such that for each EerX p i s a l i m i t point of E i f and only i f q is a l i m i t point of hCEl. Let X = { x.j , Xg, .... } and Y = { h ( ) , h(x ), Xl } 2 then for any n £ N , there i s a homeomorphism h of N * onto N * such that h ( x ) = n (x n and h ( ) x n n Mx ). n p-points)(By Theorem 3-1.3) a r e n X and Y are discrete, so there are families fK^> {HV>, N of d i s j o i n t open-closed subsets of where x e H n and h(x ) e n Vn eN. n Define IT = H^nhjj Q y . 1 Clearly x e IT and n lU^> are d i s j o i n t open-closed subsets of N and so are t n n ( u n H • By Theorem 3-1.8, there i s a homeomorphism h' of N * onto N * such that h»(U«) = h ^ I T ) h'(x ) = n n n ( x n ) = k^n)* closed set i n N N o w i f u ' i s and open- containing p, then p i s a l i m i t point of X f l U 1 and q i s a l i m i t point of h'(XnU'), hence qeh'CU ). 1 So we have h'(p)=q. .10 Corollary : For any countable i n f i n i t e set X of P-points * c of N , X has 2 l i m i t points. The set of orbits of the group of homeomorphisms of N -ft * onto N has cardinality 2°. Proof :Since any countable set X of P-points i s discrete ( Theorem 0-1.16(2)); so i t s closure i s homeomorphic to £N ( Theorem 3-1.6) and IN*| = 2 ( Theorem 0-2. C Thus, X has 2° - tf = 2 C 0 l i m i t points. There are only c permutations of X, so each u l t r a f i l t e r on X i s of the same type as at most c other u l t r a f i l t e r s on X. So there are 2° d i s t i n c t types of l i m i t points of X; then the set of orbits of the group of homeomorphisms of N has cardinality 2 . C onto N 91 CHAPTER IV HOMEOMORPHISMS OF ^3N INTO ftN 92 4-1.1 Lemma : I f f i s a mapping o f a non-empty s e t X i n t o X such that f x = x f o r no x e X, then there exist d i s j o i n t s e t s X , X , X^ such that X U X U X^ = X 1 and t'ix^l AX ± 2 = 0 1 2 1=1,2,3. Proof :Let us d e f i n e a r e l a t i o n R i n X such that for a,b e X, aRb i f and only i f f a = f b f o r some m m,n eN (J {o} n with f°a = a f o r a l l a e X. c l e a r l y R i s an equivalence r e l a t i o n and X i s p a r t i t i o n e d by R, f maps each equivalent R into i t s e l f ; c l a s s of so i t i s s u f f i c i e n t to prove the theorem f o r " a s e t X such that f o r a l l x,y eX, there e x i s t m,n e N U {o} Now choose a e X . l e a s t m e N U {o] n e N U {o} m such that f a = f x m such that f x = f ^ ^ m n m(x) a = f n(x) by n ( x ) . x x = f o r some denote the l e a s t n « N U {oj a F o r any x e X with n(x) > 0 f n For any x e X, denote the by m(x); n such that f x = f y . fn(x)-1 ( f x ) so m(fx) = m(x) and n ( f x ) = n(x) -1 so m(fx) + n ( f x ) = m(x) + n(x) -1 . For any x, e X with n(x^) = 0 that i s 93 f^Va f either m( X l = ) 1 (1) + = f°x X 1 1 a = m(f Xl ) = m(x ) +1 1 n(f ) = n( X l ) = 0 X l Therefore:m ( f ) + n ( f ) = m( ) Xl or (2) X l Xl Xl that i s 3 k( .j ) 0 £ k ( ) < m( ) + 1 m(f k( f Xl X l ) a with Xl ), = k ( = f ) + 1 such that X X l X l + 1 m ( f ) £ m( ) Xl + n( X l n(f ) ) X l ( f X i ). We claim that such an ^ i s unique : X Let x,y be d i s t i n c t points of X with the above properties. then Let b = m(x), c = m(y) „ , „o „b x = f x = f a f = f X b + 1 a = f a with b+1 >h ^ 0 and h y = f°y = f a c fy = f c + 1 a = f a with c+1 >k = 0 k without loss of generality, we may assume b > c. b >c + 1 Clearly x = f a = ff ~~ ((ff aa) = f -°y = f - " ( f y ) b bb cc cc b = f " - (f a) = f -( b c 1 k b c + b 1- ). k c 1 94 Since b > c+1 > k b > c+1 - k > 0 b >b-(c+1-k) ^ 0 . Therefore T h i s leads to a c o n t r a d i c t i o n t o the d e f i n i t i o n o f b. I f such an X 1 = e x i s t s , put ixj = 0. Otherwise Then m(fx) + n ( f x ) = m(x) + n(x) - 1 m(fx) + n ( f x ) = m(x) + n(x) + 1 Whenever or x € X - X^ Now, l e t Xp c o n s i s t o f a l l x e X - X^ with m(x) + n(x) even and X^ c o n s i s t o f a l l x eX - X^ with m(x) + n(x) odd. I t i s c l e a r that L U X , U X , = X and 1 2 fix^ nx. = 0 4-1.2 t> i = 1 ,2,3. Lemma : I f D i s a d i s c r e t e space, and f : D — » D , then the s e t o f f i x e d p o i n t s o f f , the extension of f t o a mapping o f j3D t o f3D, i s the c l o s u r e of the s e t o f f i x e d p o i n t s o f f . Proof :Let X Q be the s e t o f f i x e d p o i n t s o f f , then by Lemma 4 - 1 . 1 , 3 d i s j o i n t s e t s X^X^, X^ such that X. U X U X, = D - X 1 2 3 o 0 i = 1,2,3. and fCX.] OX, 1 1 =0^ for 95 Now c l e a r l y G l and PD o X U G 1 D 1 X P 01 p X , 01 D Q (gI) U 0 {>D2 1 X U G p l X 3 = P X , Gl p X , Gl 1 D open-closed s e t s i n D are d i s j o i n t 2 s i n c e X , X^ , X , X^ are Q 2 d i s j o i n t z e r o - s e t s i n D. Since f i s continuous and D i s dense i n (3D, the r e s u l t f o l l o w s . 4-1.3 Theorem : I f f i s a homeomorphism o f /3N onto and itself, i f P i s the s e t of a l l f i x e d p o i n t s of f , then P = p G I ( N P nN ) Proof :T h i s f o l l o w s from the f a c t that a homeomorphism of 0N onto i t s e l f i s induced by a permutation of N and Lemma 4-1.2. 4-1.4 Theorem : I f f i s a homeomorphism o f /3N i n t o N , then f has no f i x e d p o i n t . Proof :(1) F o r every k e N (i) there e x i s t s G^c N G,HG. = 0 whenever j^k, j e N and £ D k^N k = G (ii) (iii) such that k^G N k ffk> U G^ N U f CN3 . i s a neighborhood of f k i n 96 An example o f ^& keN} that : k satisfies a l l the three c o n d i t i o n s : Let us construct {F^: kcN } by i n d u c t i o n as f o l l o w s : (a) F^ We have f ( 1 ) e N * . Since J1) i s open i n pN, f ( l ) i s open i n f[/JNl therefore that there e x i s t s c N Q OfCjJN] = ff(1 )> such where Q i s Aj with A c N. 1 (b) F F : S i m i l a r l y we have f(2) e N* 2 = A - {1} . Let 1 1 Q c K * such that Q n f E^NT = (f(2)j 2 where Q 2 { f ( 2 ) } c pN - C l Let (c) F g i s A£ with A c N. 2 p N F = CI ( 1 N Clearly N - F^ ) = ( A n N - F ) - {2} . 2 2 F..: when 1 fF^:i < j} 3 Q.cl* i s defined: such that J Q. f)£lpm = J |f(j)} where Q, i s A l with A . c N. (f(d)) PN - ^ c = °V Let F j and N Cl^F. - . i ^ F i } = (AjHN - ^ 1 ^ ) - f d } • Now we have formed a sequence o f d i s j o i n t sets that i 4 F . jL {F^: i e N ) such 97 Let N - . ^ F = M. Set G i = F 1 G = F ± i G = i F i f i-1 4 M ± f " * U 1 1 i f i ' 1 € M Then we h a v e ( G : k t N ) satisfying k conditions * ( i ) and ( i i ) clearly. For ( i ) • {fk}UG i s a n e i g h b o r h o o d o f k fk i n I U fIN] : By t h e c o n s t r u c t i o n o f G , we h a v e k fk e 0 1 G 6 N 1 p ] J G G k k C G 1 P K A k <-' • U k 1 Therefore, Gl p N G n fIN] = Cl p N G k k (\ I N V f CN] ] = G Since Clp^G G (2) k U {fk} k ffk}. k i s open i n U {fk} j3N, i s open i n N 0 f IN3". F o r every n 6 G , l e t hn = f k , then h i s a k m a p p i n g o f N o n t o f CN3.. gn ^ n Set g = f ~ ^ h , then Y n e N and g maps N o n t o N. Thus g , t h e e x t e n s i o n o f g t o a m a p p i n g o f 0N o n t o j3N, h a s no f i x e d p o i n t . Since f k £C l G k i n NUfIN3 and h n = f k f o r a l l n e G , h ( f k ) = f k where h . k i s t h e e x t e n s i o n o f h t o a m a p p i n g o f PN i n t o N ; hence hx = x i f x £ f i"N3. # Now s u p p o s e t h e r e i s a n x e N f x = x. Then point of g . such that h ( f x ) = f x = x; s o x i s a f i x e d This i s a contradiction. 98 4-1.5 Theorem : If f i s a homeomorphism of (JN into i t s e l f , and i f P i s the set of a l l fixed points of f, then P = ci (pnN). p N Proof :Letx be a fixed point under f . (1 ) fCNI - N 4 0 : If fCNI - N = 0, then f l N : N —• N, and the result follows from Lemma 4-1.2. to consider the case f o r fCN1 (2) So we have - N 4 0 only. N - fINI i s i n f i n i t e : Since we have assumed that f CNI - N 4 0» there exists n. e N such that f(n.) e. {5N -N; n. i s isolated i n pN, so f(n.) i s isolated i n J J fC^NT. Therefore there i s an i n f i n i t e A<=N such that ClgjjA n f cpm = ff(n )} J Cl AnfCN] = \f(n.)} pN A Of CN] =0 and therefore N - f CN] A is infinite. s (3) xeCl^ Let , (NflfTN]) : S S M 1 = NO f CNJ, = f [NT - N, 2 = f " cs^:, 1 1 M 2 = f - 1 [S T . 2 Clearly, M A M 2 = 0 M 2 = N. j M 1 U and Thus M^, have d i s j o i n t , complementary c l o s u r e s i n |3N; so f l M ^ l , f m 2 have d i s j o i n t l c l o s u r e s i n fEjJNl. S 2 = f CM 3 C N * and 2 0N i s not homeomorphic to N , Therefore f£CT % N* p N and f . E C l ^ MgJ i s compact, so i s c l o s e d in N . Hence we have N open i n N ; so there - fCC1^ N M,,] non-empty i s an i n f i n i t e A c N such that A ' c N* - f l C l p j j M 1. 2 Since A i s countably i n f i n i t e , we can decompose A i n t o countably many d i s j o i n t infinite sets A (ne N). n The s e t s A^ are then d i s j o i n t , open-closed i n N . Now we d e f i n e a homeomorphism g o f N i n t o N such that g(n) = a i fn e^ 6 A n n g(n) = f ( n ) Now, i fn eM (g(n): n £ N } homeomorphism o f |8N i n t o N . glM 2 = flM so gloipj M 2 . infinite By Theorem 3-1.7, of g t o |8N i n t o N g has no f i x e d p o i n t . 2 i s a countable d i s c r e t e subset of N . g, the extension , isa By Lemma 4-1 .4, But 2 = f IClpjj M . 2 100 Therefore, x 4 01^ M« 2 Thus x = fx 4 0 1 p (fENI -N) which i s •incompact, so i s closed i n N , and so i s closed f t in N J (3K. We have x4 01^ (fCNI - N) , x e C l ^ (fXNl n N) , then Now we have to define a function h of N into N such that h(m) = f(m) whenever f(m) e fEN1 nN ( that i s m eM of (3) above ), 1 and h(m) £ m whenever f(m) t fINI fl N. h, the continuous extension of h to pN into £N i s a homeomorphism of 3N onto hE|5N3 and the set of fixed points of h i s the set of fixed points of f : (i) I f x i s a fixed point under f, then x = f(x). x eci^(f CNJHN) by (3), that i s f(x) € C l ^ U CNin N), so x € f ~ C l p ( f CNJ f\ N) = C l ^ f " (f INI 0 N). 1 1 N By d e f i n i t i o n , N h(m) = f(m) f(m) e fCNlfiN, whenever m ef whenever that i s h(m) = f(m) [fINIn NJ. So h o f ~ i s the identity mapping i n 1 f ENIA N, and therefore h . f " i s the 1 identity mapping i n ClpjjEf EN]ftN] . So x i s also a fixed point under h. (ii) I f y i s a fixed point under E, E(y) = y then y e C l ^ f " Cf CNJ 0 NJ : 1 f " it CN) H NI and f " If IN] - NJ are 1 1 d i s j o i n t , complementary subsets so are their closures i n 0N. of N; By theorem 4-1.2, and by d e f i n i t i o n of h that h(m) whenever f (m) 4 f i m HN, £ m whenever y eCl m e f~ 'Cf CN] fl NI, we have 1 f " If INI 0 NI = f"" Glp (fIN] 1 p N that i s 1 N As i n ( i ) above, E • f ~ ON) i s the identity mapping i n C l ^ I f EN1 fl NJ, therefore f(y) = h(y) = y. So y i s also a fixed point under f . Definition of h i n (4) : h(m) = f(m) i f m e f " [ f I N ] n N3. 1 Since N - fCNJ i s i n f i n i t e N - f " CflNIONJ 1 h(m) ((2) above). i s at most countable, l e t m be defined such that h(m)e N - f l N J ; for a l l me N - f 1 f C N l f t N J . 1 P i s the set of a l l fixed points of f . Therefore P i s the set of a l l fixed points of h. (By (2) above), and E i s the of h to a mapping of j3N to |3N. extension By Lemma 4-1.2, P i s the closure of the set of fixed points of h. By d e f i n i t i o n of h, the set of fixed points of h i s the set of a l l fixed points 102 of fIN, that i s PON. Therefore P = G l ^ ( P fl N ). 4-1.6 Corollary : If f i s a homeomorphism of ^N into i t s e l f , and i f P^ i s the set of a l l k-periodic points of f, then P k = Cl (P flN). p k Proof :Let h = f , then h i s a homeomorphism of /IN into i t s e l f ; the set of a l l k-periodic points of f i s the set of. a l l fixed points of h; therefore this corollary follows from Theorem 4-1.5. 4-1.7 Remark : Theorems 4-1.3 and 4-1.4 of Theorem 4-1.5. are particular cases 103 BIBLIOGRAPHY [1] N. Boboc and GH. S i r e t c h i , Sur l a compactification d^un espace topologique. B u l l . Math. Soc. Math. Phys. R. P. Roumaine (N. S.) 5(53) [2] (1961) 155-65(1964). Z. F r o l i k , Fixed points of maps of /8N. B u l l . Amer. Math. Soc. 74(1968), 187-191. [3] L. Gillman, The Space |3N and the Continuum Hypothesis, General Topology and i t s Relations to Modern Analysis II, second Prague Topological Symposium (1966) 144-146. [4] L. Gillman and M. Jerison, Rings of continuous Functions. Van Nostrand Princeton, I960. [5J M. Katetov, A Theorem on Mappings. Comment Math. Univ. Carolina [6J 8(1967), 431-433. J . Kelley, General Topology. Van Nostrand N.Y. (1961). [7] K. D. Magill, J r . , The Lattice of Compactifications of a Locally Compact Space. Proc. London Math. Soc. (3) 18(1968) 231-244. [8] I. I. Parovicenko, On a Universal Bicompact of Weight 104 M. Soviet Mathematics Doklady 4 148-150(1963) 592-595. [9] M. E. Rudin, Types of U l t r a f i l t e r s . Topology seminar Wisconsin, 1965, Annal of Mathematics Studies 60, Princeton University Press, Princeton (1966) 147-151. [10] W. Rudin, Homogeneity Problem i n the Theory of Cech Compactification. Duke Math. J . 23(1965), 409-420.
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Homeomorphisms of Stone-Čech compactifications Ng, Ying 1970
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Title | Homeomorphisms of Stone-Čech compactifications |
Creator |
Ng, Ying |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | The set of all compactifications, K(X) of a locally compact, non-compact space X form a complete lattice with βX, the Stone-Čech compactification of X as its largest element, and αX, the one-point compactification of X as its smallest element. For any two locally compact, non-compact spaces X,Y, the lattices K(X), K(Y) are isomorphic if and only if βX - X and βY - Y are homeomorphic. βN is the Stone-Čech compactification of the countable infinite discrete space N. There is an isomorphism between the group of all homeomorphisms of βN and the group of all permutations of N; so βN has c homeomorphisms. The space N* =βN - N has 2c homeomorphisms. The cardinality of the set of orbits of the group of homeomorphisms of N* onto N* is 2c . If f is a homeomorphism of βN into itself, then Pk , the set of all k-periodic points of f is the closure of PkՈN in βN. |
Subject |
Stone-Čech compactification. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-06-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080509 |
URI | http://hdl.handle.net/2429/35352 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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