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Homeomorphisms of Stone-Čech compactifications Ng, Ying 1970

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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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