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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 University, 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f M gjk-P,MxaXx c ^  T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e Water /£, es y ABSTRACT The set of a l l compactifications, 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 with |3X, the Stone-Cech compactification of X as i t s largest element, and flX, the one-point compactification of X as i t s smallest element. For 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 compactification of the countable i n f i n i t e discrete space N. There i s an i s o -morphism between the group of a l l homeomorphisms of and the group of a l l permutations of N; so |3N has c homeo-r,. * c morphisms. The space N =|3N - N has 2 homeomorphisms. The ca r d i n a l i t y of the set of orbits of the group of homeo-* * c morphisms of N onto N i s 2 . I f f i s a homeomorphism of into i t s e l f , then , the set of a l l k-periodic points of f i s the closure of P , n N i n |3N. i i i ACKNOWLEDGEMENTS I am deeply indebted to Professor J. V. Whittaker for suggesting the topic of this thesis and for rendering invaluable assistance and encouragement throughout the course of my work. I would like to thank Professor T. E. Cramer for reading the fi n a l form of this work. I gratefully acknowledge the financial support of the Department of Mathmatics of the University of British Columbia. i v TABLE OF CONTENTS Page INTRODUCTION 1 CHAPTER 0 : Preliminaries 2 CHAPTER I : A Theorem on homeomorphisms of j3X - X and j3Y - Y, with X and Y l o c a l l y compact, non-compact Hausdorff spaces 27 CHAPTER II: Homeomorphisms of j8N 67 CHAPTER I I I : Homeomorphisms of N* 73 CHAPTER IV: Homeomorphisms of |SN into |3N 91 BIBLIOGRAPHY 103 INTRODUCTION For any locally compact, non-compact Haus-dorff space X, at least we have |3X, the Stone-Cech compactification and AX, the one-point compactifica-tion. In chapter I, we study K(X), the set of a l l compactifications of X, a locally compact, non-compact Hausdorff space, and partially order i t by defining cx^ X ^ oCjX i f there exists a continuous function map-ping o^X onto o<jX which leaves the points of X fixed. K(X) is a complete lattice under such an ordering. We consider (3X, ^Y, the Stone-Cech compactif ications of X, Y and find out the relation between K(X) and K(Y) when there is a homeomorphism from |3X - X onto pY - Y. In chapter II, 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 in chapter III and the homeomorphisms of |3N into |3N 4--( including into N ) in chapter IV. The property * df Du-Bois-Reymond separability of N plays an imp-ortant role in the proofs of theorems concerning homeomorphisms of N onto N and those of ^ SN into p . CHAPTER 0 2 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, definitions, and fundamental results required by or used throughout this thesis are summarized in this chapter. This chapter i s divided into four sections. In section 1, the fundamentals for 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 funct-ions from X to R. (3) Z(f) ={x € X : f(x) = 0 } , the zero-set of f , is the set of a l l elements of X whose image is 0, where f £ C(X). (4) Any set that i s a zero-set of some function in C(X) i s called a zero-set in X. (5) Z(X) ={z(f): f«C(X)} is the family of a l l zero-sets in X. 0-1.2 Theorems : (1) Every zero-set i s a G^  and is closed. (2) Z(X) is closed under countable intersections. 0-1.3 Definitions : (1) Two subsets A and B of X are completely separated in X i f there exists a function f € C (X) such that f [A)= {o}, f[B] = {l] and O ^ f ^ l . (2) A subspace S of X is C -embedded [C-embedded]in X i f every function in C (S) [C(S)J can be extended to a function in C (X) [c(X)] . 0-1.4 Theorems : (l) Two subsets of X are completely separated i f and only i f they are contained in disjoint zero-sets in X. (2) Urysohn's extension theorem : A subspace S of X * is G -embedded in X i f and only i f any two com-? pletely separated sets in S are completely separated in 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 ( i i ) i f Z l t Z 2 € J.then Z^ 3F; and ( i i i ) i f ZeJ, Z* £Z(X), and Z ' D Z , then z'e J . (2) Every family (8 of zero-sets that has the fi n i t e intersection property i s contained in 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-fi l t e r ? . When (6 i t s e l f i s closed under fi n i t e intersection, i t is called a base for J . ( 3 ) By a Z-ultrafilter on X is meant a maximal Z-f i l t e r , i . e . one not properly contained in any other Z - f i l t e r . Thus a Z-ultrafilter i s a maximal subfamily of Z(X) with the fi n i t e inter-section property. 5 0-1.6 Theorem : If , Q> are d i s t i n c t u l t r a f i l t e r s on X, tnen there exists EcX s.t. E«I2j and X - Ee.Q^ . 0-1.7 Remark : In a discrete space, every set i s a zero-set, so that f i l t e r s and Z - f i l t e r s ; u l t r a f i l t e r s and Z - u l t r a f i l t e r s are the same. 0-1.8 Definitions : (1) A space X i s said to he completely re-gular provided that i t i s a Hausdorff space such that, whenever F i s a closed set i n X and xeX - F, F and jx} are comp-l e t e l y separated. (2) A c o l l e c t i o n (8 of closed sets i n X i s a base fo r the dosed sets i f every closed set i n X i s an intersection of members of (ft. Equivalently, (B i s a base of closed sets i f whenever F i s closed and xeX - F, there i s a member of (8 that contains F but not x. (3) A family (8 of closed sets i n X i s a sub-base fo r the closed sets i f the f i n i t e unions of i t s members constitute a base for the closed sets. 0-1.9 Theorems : (1) Every closed set in a completely regular space is an intersection of zero-sets. (2) A Hausdorff space is 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 disjoint closed sets, one of which is compact, are completely sep-arated. (b) In a completly regular space, every G^  containing a compact set S contains a zero-set containing S. Hence, every compact is a zero-set. 0-1.10 Theorems : (1) (a) Every subspace of a completely regular space is 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) If (B is a base C subbase ] for the closed sets in X, then { A = X - B : Be©) forms a base I subbase 1 for the open sets in X. 7 (3) (a) Every closed set F in a completely regular space i s an intersection of zero-set-neighborhoods of F. (b) Every neighborhood of a point in a completely regular space contains a zero-set neighborhood of the point. 0 - 1 . 1 1 Definition : Let X be a completely regular space. A point p € X i s said to be a cluster point of a Z-filter 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 i f p e ^ e Z - f i l t e r $ i s said to converge to the limit p i f every neighborhood of p contains a member of J . 0 - 1 . 1 2 Theorems : 3F i s a Z- f i l t e r on a completely regular space X, p € X: ( 1 ) (a) If J" converges to p, then p i s a cluster point of 5 . (b) A Z-ultrafilter 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) If p is a cluster point of J .then at least one Z-ultrafilter 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-ultrafilter converging to p. (c) Distinct Z-ultrafliters 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-ultrafilter containing J . 0-1.13 Definition : A hausdorff space i s said to be locally 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) If T i s locally compact, and X is open in T, then X i s locally compact. (b) If X i s dense in T, then every compact neighborhood in X of a point p f X i s a neighborhood in T of p. (c) If X i s dense in T and p i s an isolated 9 point of X, then p is isolated in T. (d) If X i s locally compact and dense in T, then X is open in T. 0-1 .-15 Definition : A point p in a topological space X is a P-point of X i f every countable intersection of neighborhoods of p contains a neighborhood of p. 0-1.16 Theorems : (1) If X is completely regular, then p is a P-point of X i f and only i f for 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 in a Hausdorff space is discrete. 0-1.17 Definitions : A family of subsets of a space X is a ring i f i t i s closed with respect to finite unions, finite intersections and complementations. 0-1.18 Theorem : Each countable family of sets i s contained in a countable ring. 0-1.19 Definition : 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 this 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 in which X is 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 is dense in a space T. (1) If J is a Z-fi l t e r on X, then p e T i s a cluster point of J provided that P « Z & C 1T Z (2) If Z i s a zero-set in X, and p e Cl^Z, then at least one Z-ultrafilter on X contains Z and converges to p. (3) Every point of T is the limit of at least one Z-ultrafilter on X. 0-2.4 Theorem : Every completely regular space X has a Stone-fiech 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 from pX into Y. ( 2 ) (Stone-Cech) Every function f in C * ( X ) has an extension to a function in C(|3X). (3) (dech) Any two disjoint zero-sets in X have disjoint closures in p X . ( 4 ) For any two zero-sets and Z 2 in X , ( 5 ) Distinct Z-ultrafilters on X have distinct limits in |3X. * (6) If X i s dense and C -embedded in T, then X.crT< (7) If X i s dense and C -embedded in T, then |3T = p X . Furthermore, p X i s unique, in the following sense: i f a compactification T of X satisfies any one of the listed conditions, then there exists a homeo-morphism 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 limit points of Z-ultrafilters on X . So the family of a l l Z-ultrafilters on X i s written ( A p ) p e ^ where A p i s the Z-ultrafilter on X with limit point p. 12 (b) Define Z pe[3X : Z <£ Apj f o r Z e Z ( X ) . J3X i s made into a topological space by taking the family of a l l sets 2 as a base for the closed sets. C1.TZ = 2 . (c) p e Cl f l TZ i f and only i f Z e A p. Every point of |3X i s the limit of a z-ultra-f i l t e r on X. So there i s a natural correspond-ence of elements of X^ and Z-ultrafilters on X. Corresponding to 2 = {p e J3X : Z e Ap} we have Z = { n : Z £ Ii, £2 is a Z-ultrafilter on X} for Ze Z(X), when we consider elements of (SX as Z-ultrafilters on X. 0-2.6 Theorems : S i s a subspace of X, (1) S is C*-embedded in X i f and only i f c i s = p s (2) Every compact set in X is C -embedded in X. (3) If S i s open-closed in X, then C1.,YS and Clpy^X - S) are complementary open sets in |3X. (4) An isolated point of X i s isolated in j}X, and X is open in |3X i f and only i f X i s locally compact. 0-2.7 Theorem : For any infinite discrete space Where IXI i s the cardinality of X. 14 Section III The set N of a l l positive integers i s chosen as a countable discrete space. Since N i s locally compact, N i s open in p N . Every point of N is an isolated point of p N . These are the only isolated points of p N , since N is dense in p N . Hence every fin i t e subset of N i s open-closed in p N . Since every subset of N i s open-closed in N and i s also a zero-set so the closure in p N of every subset of N i s open in p N . 0-3.1 .Theorem : Every open-closed set in |6N i s of the form Clp^A for some A c l . So there are c open-closed sets in pK, where c i s the continuum cardinality. Proof :-Suppose Sep i s open-closed in p u , then B = pN - A" i s also open-closed in jSN. Let A = S D N , B = B/1H. Clearly I = Cl^A, i f not, ClpjjA A" (by definition, 01 ^k is the smallest closed set in p N that contains A). Let A' = A - C l ^ . which is non-empty open-closed in ^QN. Since N is dense in p N , A ' r i N £ 0. This i s a contradiction. 0-3.2 Remark : It follows from the construction of p x that •A 15 the open-closed sets of AN form a base for the closed sets of AN. Since the complement of every open-closed set of 0N i s also open-closed i n -3N, so they also form a base f o r the open sets of AN. (by Theorem 0-1.10(2)). 16 Section IV # Now we consider the subspace N = pN - N of j@N. Since (3N i s compact and N is open in j8N, so N is closed in pN and therefore N i s compact perfect. For Ac:N, define A' = Gl^A - N = C l ^ A fiN*. Clearly, A i s fi n i t e i f and only i f A'= 0, and A'cB' i f and only i f A - B i s f i n i t e ; so A' = B1 i f and only i f ( A - B ) U ( B - A ) i s f i n i t e . It follows that there are c distinct sets A'.(c i s the continuum cardinality.) Since {Cl^A : AcN forms a basis for the open sets of (JN and also a basis for the closed sets of |3N, so does {A1 = ClpjjArtN*: A C N ( for N* . 0-4*1 Theorem : * Every open-closed subset of N i s of the form A' = C l ^ A - N = Cl^AHN* for some AcN. Proof :-Suppose W i s open-closed in I , V = N - W is also open-closed in N . Since p N , N 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 : f (n) > £}, C= N - A - B = f n e N : f (n) = i ) . Since jx €pN : f(x) = ±) nN* = 0 and {xe |SN : f(x) = *} is closed in |?N, we have Cl p C fl N = 0 . .'. C i s f i n i t e ; and ¥ = A* = C l ^ A fiN*. 17 0-4.2 Theorem : The intersection of any countable family of open subsets of N is either empty or contains a non-empty open set. Proof :-Let {G^ } be a countable family of open subsets of N whose intersection contains a point a. Since [A' = Cl^A^N* : AcNJ forms a basis for the open sets, there are infinite sets A^c N s.t. aeA|c:G i V i e N. The intersection of any finite collection of the sets A^ i s non-empty and open. For F, any fi n i t e collection of the sets A^, there corresponds F*, a finite, collection of the sets A^. Since Pi A ! ^ 0 and AjeF* x A ^ V i = C 1 F N A9F A1 (Theorem 0-2.4(4)), AOT-»A. 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 fi n i t e for each i € N 18 because zi^A-^ whenever j ^  i , A - A. c f i i | , ... n ^ ] ; so A'<-=A£ VieN. " A c i e N A i ieN u i A is infinite, .*. A» ji 0 and A' i s open-closed in N• . 0-4.3 Definitions : If S is a set partially ordered by -< , e, h, e.j , e2> •••••» ©n> h^, l ^ . , h^, , e.|, C2» ••••»cn, .... are in S, then S i s said to have the properties : (1) Simplest separability , i f for each e< h, there exists a geS s.t. e<g-<h; (2) Separability of Cantor, i f for any set e.j< ... <en 4 ^ h of type w+ 1 , there exists age.S s.t. en-<g-<h Vn £ N; and (3) Separability of Du-Bois-Reymond, i f for each set e 1 < e^< ....-< e n < .... < h < .. .. < hp < h.j of type w + , and any set c 1, c 2, ...s.t. h n ^  c ± and e Q for any n and i e N , there exists a geS s.t. e n<g<h n V n €N, and c±^ g ^  o± for any i £ N. 19 0-4.4 Theorem : Let LN be the family of a l l open-closed # -si-subsets of N , a basis of N , with the p a r t i a l order of set inclusion, LN has the properties of simplest separability, separability of Cantor and separability of Du-Bois-Reymond. Proof :-* (1; LN i s simplest separable : Let A» f B» e LN*, 3 y e ( B' - A' ) € LN*, Since N i s perfect, H y ^ Z e l B ' - r A ' ) ( B 1 - A' - Z ) i s an open but not closed neighborhood of y. . \ 3 0 j t C ' e LN* s.t. y € C'c B f - A' - Z A'^C'UA'^t B' - A* )UA' = B' 12) LN i s Cantor separable : Let Aj g A£ £... ^  £ ... fF B' where B« , A| €LN* V i eK, Put C| =• N* - A| V i e N and D1 = N* - B*. Clearly 0^ e LN* V i e N , D' e LN*, and ^ * '''' * n ¥ * * Set E£ = G*± - D' . Ej_ e M* V i e N, and By Theorem 0-4.2, we have j^EJj^ containing an H' s.t. H* 6 LN*, and Let K1 = N* - ( H'U D1 ), then * LN i s Du-Bois-Reymond separable : Let Aj_ and B| be i n LN* V i e N s.t. -D1 4: •••• - ° n * •••• ^ + A-| • with A£ = C1^ N A. A N* ( B i = 0 1 BN B i n N * ' A i ' B i i 1 1 - ^ 1 1 1 1 ^ 6 subsets of N, V i eN, and B1 < B 2 < < B n < ^ A n< ....-< A2-< A1 with •< a p a r t i a l order defined by A< B i f and only i f A - B i s f i n i t e and B - A i s i n f i n i t e . Let C*, G^, .... be i n LN s A'cfcC! and n T x C ^ B ^ V i , ne N, G i = G 1|3H Gj_^ N*» c i an i n f i n i t e subset of N. i . e . ° i * A n ^ C i ^ B n o r A,, - C. and C. - B_ are i n f i n i t e for n 1 i n a l l i , n 6 N. 21 Now construct E^,K^ by induction as follows : H1 = U j H B,) \J{VU] where p 1 1 £ A 1 - ^ K 1 = Unl where *n 6 °i ~ H i can be chosen in C^- because H 1CB 1 UfP^I and C1 - H1 U {P 1 1)r>C 1 - B1 i n f i n i t e . Suppose H1 , H2, ...., Hn_1 , , K2, ...., are constructed, with f i n i t e Vi = n-1. Let t n be an integer greater than any member of K1 (J K 2 U .... U Kn_1 which is f i n i t e , and L = k.(\ k0f\ f)k . n 1 2 n H n = { m ( I n f l B n : m > ^  } U { p^.ito } where p. eL - C. and p. > t. . for i = n. in n l m n The choice of p i n i s possible since A n - CL is infinite and A - L = A - A, 0 A 0 H f\k n n n 1 2 n n iWl(An " A i ^ i s f i n i t e since A < A ^ , , < . . . . • < A 0 < A „ , A ^ - A . is n n—1 2 I n i finite for a l l i £ n. H n has the following properties : (i) H n i s infinite : L n B = A . n k0 n — n A r\ B n n 1 2 n n = ( A 1 n B n) n u 2 n B n) n.... n ( A n n B N ) For 1 = 1 , 2 , ....,n A^ i s infinite B N - A ± is fi n i t e ( V B N - < A ± ) 22 13 i s i n f i n i t e n .'. (A if\ B n ) i s i n f i n i t e , and n iQl ^ i 0 2 ^ ' = G n ( T h e o r e m s 0-4.1, 0-4.2) n with G n = S)^ {A±f) B n) i n f i n i t e (Theorem 0-2.4(4)). H n D G n - }m : m<t n) 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 H n C L n n B n U ( Pin : 1 * n ) H n C B n U { P i n : .*. H - B„ i s f i n i t e n n •'• H i " BnC - Hi " V " <Bi - V which i s f i n i t e V i ^  n since B ± - B n i s f i n i t e V i £ n. (Iii) B n - H n i s f i n i t e : B n " H n C ^ B n " ^ L n n B n ) U ( m £ L n n B n : m ^  t n ] and Bn" • LnnBn > = Bn " < A 1 ^ • • • > = iW,< Bn - V B„ - A. i s f i n i t e V i e N. n x .*. B^ - H i s f i n i t e , n n (iV) H n - A k i s f i n i t e Vk € N : H n C B n U { P i n : i ~ n ) a n d B - A i s f i n i t e Vk € N n k .*. H n - A k i s f i n i t e Vk e IT. 23 (V) Hn< A i Vi € N H n C B n ^ f P i n : i * n ) P> - A. fin i t e Vi e N n 1 .'. H - A, f i n i t e Vi e N n 1 A. - B„ is infinite Vi eN l n A. - H i s infinite Vi e N. x n .'. H < A, Vi eN. n l (Vi) H nC k± Vi = n H nc L n = A1 n A 2 D . . . . n A n^ k± Vi ^  n Define K n = { q ± n « 0 i - J ^ : qi n> V i = nj. The choice of q ^ n is possible because n C i - B k infinite Vi, k e l , and ^ \ - B n is f i n i t e by property (ii) of HR. C. - B„ i s infinite Vi, n eN. i n n C. - , U H, i s inf i n i t e , l n=1 h Put D = 5.H n=1 n For any fixed k we have D = P^U Qk where P k = H1 U H 2 U U H k Q k = H k + 1 U H k + 2 U We have shown that B - H is fi n i t e n n (property (iii) of H n ) .'. B k - P k is fin i t e 24 B k - D i s f i n i t e D " P k = ^ +1 U Hk+2 U '' *' ^  ^+1 i n f i n i t e . .*. B 4 D Also Q k ^ A k + 1 V \ = EMU\+2U — • C Ak +1 ( by property (Vi) of H n ) •'• Q k ~ Ak+1 = $ P k - M H j < Ak+1 V j € N ( property (Vi) ) .. P k - A k + 1 i s f i n i t e . •'• D - A k + 1 = < P k - A k + 1 ) U ( Q k ~ W i s f i n i t e . A k + 1 < A k given A k - A k + 1 i s i n f i n i t e A k - D = A k - ( D f U k + 1 ) - ( D - A k + 1 ) => A k " Ak-M " ( D " Ak +1 } i S i n f i n i t e . D < A k Hence we have B, -< B < ... < B < ... <J>< ... <A < ... < k. 1 2 n n 1 25 For any fixed CL, there are inf i n i t e l y many p. «D, p. 4 C., so D - C. is in f i n i t e . The choice of the number q i n shows that 0 . - D i s inf i n i t e . Thus D* has no set x inclusion relation with 0/ Vi e N. 0-4.5 Definition : In 0 - 4 . 4 ( 3 ) , i f 3 M' € LN* s.t. M' (\ Aj_ £ 0, M'0 B| = 0 V i e N , then we can have D' disjoint 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 ( By Theorem 0-4.2 ). If in addition P' = M» H J^^ A| is open-closed in N , we may choose D' s.t. P'C D* by choosing D' as the union of the D1 described in Theorem 0-4.4 and P'. Let this property be called the extended Du-Bois-Reymond separability. 0-4.6 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 partial 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 clearly shown in the proof of Theorem 0-4.4(1). •=->• is true, for i f x is an isolated point, 3 no B s.t. 0 fi B ^ {x}.) 0-4.8 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 given spaces and a l l compactifications are Hausdorff and a l l given l o c a l l y compact spaces are non-compact. 28 Section I : In t h i s section we study the set of a l l compactifications of a l o c a l l y compact, non-compact, Hausdorff space. For brevity's sake, i n t h i s section we frequently use the alternative d e f i n i t i o n of compacti-f i c a t i o n : a compactification of a topological space X i s a pair ( o<X, h ) consisting of a compact Hausdorff o(X and a homeomorphism h of X onto a dense subset of oCX. Two compactifications c<X, TX of X are equal i f there exist a homeomorphism (p of c*X to TX such that i f h ^ x ) = h y(x) Vx eX. I f ( (XX, h^) i s a compactif i c a t i o n of X, c<X - h dCXI = 3(X, oCX) i s the boundary of X r e l a t i v e to <*X. 1-1.1 D e f i n i t i o n : l e t X be a topological space, T a compact space. A family J(X, T) of continuous functions from X into T i s completely regular i f f o r every closed FcrX and x Q 4 F, there exists f e ^ C X , T) such that f ( x Q ) 4 fTFT . 29 1-1.2 Theorem : Let X he Hausdorff, T compact,? a completely-regular 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. C f(hj(x)) = f(x) Yx eX 3 . (b) If y', y" e 7X, y» £ y" then there exists V l i M f e J s.t. Proof :-(1) There exists such a compactification of X : Consider the compact Hausdorff space * fejF x f where T^ = T whenever f e J, and the mapping q : X —*• E defined by 7i f(q(x)) = f(x) Vx € X where TC^  i s the projection 7 l f : E T f. If f e J, then 71^,* q i s continuous and, so is q. Since X i s Hausdorff, J i s completely regular, for any x', x"€ X, x 1 £ x " , there exists f € 7 s.t. f(x f) £ f(x"), hence Tt f(qU')) * n f U U " ) ) i . e . q i s one-to-one. Let G be open i n X and X q e G. Then X - G i s closed and x Q 4 X - G. J i s completely regular, so there exists f € ? s . t . f ( x Q ) * fCX - G] , .'. there exists YQ, an open neighborhood of f ( x Q ) s.t. V„ n f CX - G] = 0 . o f i s continuous, .'. x f~^(V 0) i s open i n E, and we have q ^ ) e q(X )n7U f"" 1(V 0)cq(G) .*. 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 defined by : hj(x) = q(x) V x eX. Clearly, (3FX, h^) i s a compactif i c a t i o n of X. We have the diagram : where f i s restricted to ?X. If y', y"e 7X, y' £ y" there exists f € 3 s. t. TTfCy') ^ * f ( y " ) .V f(y') ^  f(y") So (JX, hj) i s a compactification of X that satisfies (a) and (b) above. (2) The compactification (JX,h?) i s unique: Let (<*X, h^) be another compactif ication 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' is 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 32 (pChrfCx)) = hy(x) for a l l x e X by definition of 4> ; and hy(X) i s dense in JX. Therefore f ( dX) = J(X) Therefore f is a homeomorphism of o(X onto ?X, with <f(h^(x)) = hj(x) . Therefore tfX = JX. 1-1.3 Corollary : A space i s compactifiable ( has a compactificat-ion ) i f and only i f i t is 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 is practicable i f there exists a continuous function which is not constant from [0,11 into T. 1-1.5 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 ication of X. Since T is practicable, there exists a continuous if which is not constant such that 33 if maps CO, 11 t o T. Without 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. L e t he a c l o s e d s e t i n X and x e X - F „ , 1 o 1 t h e n t h e r e e x i s t s F 2 c l o s e d i n e<X such t h a t x- e c<X - F 2 and F^ = F , , n X. S i n c e o<X i s compact, t h e n normal, t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n f t h a t maps o(X t o 1 0 , 11 such t h a t f ( x o ) = 0 f C F 2 I = {1} T h e r e f o r e f(0) = <f ( f ( x Q ) )^  <f* ( f CF1 ) = (f(l) i n T. S i n c e T i s H a u s d o r f f , n o ) ^ w r y t h a t i s <Mf(xo^ ^ vTfCFpT <f f ( x Q ) ^ <ff CF21 . T h e r e f o r e f ^ l x 1 S a f u n c t i o n from X i n t o T such t h a t (a) < P f l x ( x 0 ) 4 ^ f l x ( F 1 ) and (b) ^ f l ^ has a c o n t i n u o u s e x t e n s i o n t o (*X. Now l e t 3^  be the s e t o f a l l c o n t i n u o u s f u n c t i o n s o f X i n t o T w i t h c o n t i n u o u s e x t e n s i o n t o <XX. Then \ i s u n i q u e l y determined by (XX and T, and i s a c o m p l e t e l y r e g u l a r f a m i l y . I t i s a l s o c l e a r t h a t f o r y', y" i n dX, y", t h e r e e x i s t s f e % s o t h a t f y ' * f y " . 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 functions associated with o<X. Then o<X = X. Proof :-The compactif i c a t i o n (?<X s a t i s f i e s both condit-ions (a) and (b) of Theorem 1-1.2 r e l a t i v e to the family JU . Therefore the two compactifications are equal. 1-1.7 D e f i n i t i o n : Let X be compactifiable and K(X) be the set of a l l compactifications of X. For two elements <XX, YX of K(X), c<X = yX i f there exists a continuous function ¥ from tfX onto oCX such that »f(hy(x)) = h^ x ) for a l l x g X. Since h< CXI and h IX] are dense subsets of (A X and TX respectively, y and ip i s continuous, so <f EoCXI i s compact. Thus ip i s uniquely determined. Clearly the r e l a t i o n =" i s a p a r t i a l order i n K(X). From theorem 0-2.4, we have |3X the largest element of K(X). 35 1-1.8 Lemma : Let X be compactifiable, <?(X, YX be two compact i f icat ions of X such that <XX = YX . Then the function if that maps T X to o<X induced by the relation =" satisfies the condition. f c 9 u , r x ) a = a (x , <-<x) that i s <f C T X - Xi = o(X - X. Proof :-(1) Clearly if E Y X - X I => cCX - X, since <p i s onto and i t leaves X invariant. (2) f t tX - X I c oCX - X. If ye YX - X such that cp(y) ^ <*X - X, then clearly <f(y) = x Q for some x Q€ X c o(X and *f (x Q) = x Q. Since x Q ^ y in Y X , we can choose open neighborhoods V of x Q and U of y in YX such that V n U = 0; then Yf\ X i s open in X, and there exists open neighborhood of X Q in o(X such that V O X = X. Since f is continuous, for in o(X there is a neighbor-hood VY of y such that V TCU in Y X and if I V Y ] C V1 in <*X. On the other hand, X i s dense in Y X , there exists x^  e X O Y T ' ^ U then clearly x^  ^ V because U f l V = 0, and x 1 = if (x^ ) e V 1 c o(X. Therefore x ^ ^ H X = VOXcV. This is a contradiction. 36 1-1 ..3 Theorem : Le t X be a t o p o l o g i c a l space, T p r a c t i c a b l e , J , » J T two c o m p l e t e l y r e g u l a r f a m i l i e s o f c o n t i n u o u s f u n c t i o n s of X i n t o T. I f ^ c ? r , t h e n dX ^ YX where <<X, JX are the c o m p a c t i f i c a t i o n s o f X a s s o -c i a t e d w i t h the f a m i l i e s 3* , 3y r e s p e c t i v e l y . P r o o f :-T h e r e f o r e every f e ^ can be extended t o a c o n t i n u o u s f o f *X. L e t i f , a map from KX t o o(X be d e f i n e d by : < p ( y ) = { ^ y ) J f € 3 P - . C l e a r l y «flhj(x)l = h^Cx) t h a t means the f o l l o w i n g diagram i s commutative : x 1 Y x E v i d e n t l y u» i s c o n t i n u o u s , and TJX] i s compact i n <XX, «pEhT(x)I i s dense i n o(X; t h e r e f o r e o(X = (? I I X J o r ^ i s a c o n t i n u o u s f u n c t i o n from YX onto o(X. so o(x ^ j r x . 37 1-1.10 Theorem : Let p(X, fX be two compactif ications of X. Then o(X ^ n 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, for T practicable. Proof :-(1) Necessity : <<X ± XX ; by definition, 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 is a continuous extension of f to XX. (2) Sufficiency : Let , J y be the two completely regular families of continuous functions associated with -XX, ?X respectively. By hypothesis c. 3y ; and by theorems 1-1.6 and 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. By Theorem 1-1 .6 , ^ X = o(X. Let 3- = JJft and ST-X be the compactif ication of X associated with J. By Theorem 1-1.9, <*X = J X . If 9 X i s a compactification of X such that o(X * 9 X for 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 Q Therefore 3 X £ JQX = 9 X .. Therefore i X = sup <AX. 39 1-1.12 Theorem : If K(X) is the set of a l l compactifications of a locally compact space X, then every non-void subset of K(X) has an infemum. Proof:-Since X i s locally compact, (ZX, the one-point compactification of X i s obviously the smallest element of K(X). Let {o(X} ^ e A c K(X). If a X € { o f X ) o ( £ A then ax i s the infemum. If ax 4 {oCX}^ and i t does not contain a minimum (else i t is t r i v i a l ) then by Theorem 1-1.11, j }fX : yX < o(X V *<€ A} has a supremum X^X. Then Y^X is the infemum of |<^X) ^ e A« 1-1.13 Corollary : If X is locally compact then K(X), the set of a l l compactifications of X forms a complete latt i c e . 40 Section 2 : In the last section we have defined on K ( X ) , the set of a l l compactifications of X the partial order of =". Por any compactif ication 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^ is unique, and this function w i l l be referred to as the 0-function of erfX in this section. By Lemma 1 - 1 . 8 , i t is obvious that the p-function f^ induces a decomposition of | 3 X - X into a family 7 (e<X) of mutually disjoint non-empty closed subsets where 7 ( c * X ) = { f~^ (p) : p e c<X - XL In this section, ^ ( t ? < X ) w i l l be referred to as the /5-family of e < X . 1-2 .1 Lemma : Let c<X , T X be two compactif ications of X . Then o C X = T X i f and only i f each set in J ( y X ) i s a subset of a set in 3( o < X ) . Proof : -( 1 ) Necessity : If o<X = y X , then by definition, there exists a continuous function h from f X onto o<X such that h(x) = x fom a l l x e X . Let f ^ and f r be the ft-functions of o(. X and T X respectively. Then h . f r(x) = h (x) = x = f ^ x ) for a l l x e X . So, h« f y agrees with f^ on X , a dense subset 41 of px. Since any two continuous functions from an arbitrary space into a Hausdorff space must be i d e n t i c a l i f they agree on a dense sub-space, thus h»f y = f^ . Now l e t A be any set i n J ( JX), then A = fj" 1(p) for some p e 7 X - X. Let q e A, then f r(q,)= p and h(p) = h*f r (q) = f„<(q) e oCX. So we have qe f j 1 ( h ( p ) ) and hence AcrfJ 1(h(p)) a member of 3(o(X). ( 2 ) Sufficiency : Let f^ , f r be the p-functions of c<X, XX respectively, p be a point i n YX - X. By hypothesis, there exists a unique point q i n X X - X so that f 7 1 (p) S f , 7 1 ( q ) . Define a mapping h from Y X to <AX by h(p) = q and h(x) = x fo r a l l x € X. Suppose A e pX - X, then f 7 1 ( f Y ( i ) ) £ f , 7 1 ( t ) for some t e *tX - X and h(f T U ) ) . = t. Moreover, A € f ~ 1 (f,(A)) c f " 1 ( t ) , f^U) = t, therefore h o f f ( f t ) = f^U) and h«f r(x) = h(x) = x = f K ( x ) f o r a l l xe X. Hence we have f^  = h o f . Clearly h i s onto. Let K be closed i n <XX, then f~^ (K) i s closed i n px, so i s compact; and therefore fyCf^  (K)) i s compact, then closed i n YX. Since f ^ = h»f j< and h " 1 = f y ' f * 1 we have h' 1(K) = f r ( f ~ 1 ( K ) ) closed i n YX. Thus h i s continuous. So we have P<X ^ XX. 42 1-2.2 Remark : Lemma 1-2.1 implies that a compactification is uniquely determined' by i t s p-family. 1-2.3 Lemma : Let X be locally compact, o(X a compactif ication of X, , Kg,..., KJJ be N mutually disjoint non-empty closed subsetsoof c<X - X. Choose N distinct points q^  , q 2».. • ,q N not in o<X and define a mapping h from o<X onto TX = C <*X - J ^ I L I U {q±: 1=1, ,NJ by h(p) = p for a l l pe MX - J^K^ and h(p) = q ± for p e K^. Let TX have the quotient topology induced by h,that is the largest topology on TX such that h i s continuous. Then TX is a compacti-fication of X. Proof :-Since o<X is compact, TX has the quotient topology induced by h and hCcrfX] = TX, and X is dense in <XX, we have TX compact and X dense in TX. Now we have to prove that TX is Hausdorff: Let G = (XX - jJLjK^ • G is open in e(X, and each i s closed in y x . G = TX - { q ^ i=1 ,... ,N) so i t is sufficient to consider the following three cases for distinct points p and q of TX in order to prove that YX is Hausdorff. 43 (1 ) P = .qm and q = q n ( 2 ) p € G and q = q m (3) p, qeG. Case (1) ,:•Since each is closed in ©<X, which is compact, there exist disjoint open subsets U and U of o<X such that m n K m c U , K c U , U n K. = 0 for m m n n m l i ^ m and U nH K ± = 0 for i ^ n. Let U* = h(U ) and U* = h(U ). m v m n x n Both U.. and U are open in YX m n since U m = h _ 1(U*), U n = h"1(U*) and YX has the quotient topology induced by h. Clearly U m and U are disjoint neighborhoods of p and q respectively. Case ( 2 ) : There exist disjoint open subsets U and V of c<X such that p e TJ, K G V, ' m ' U H L = 0 for a l l i and V 0 K±= 0 for i £ m. Let U*= h(U) and V*= h(V). Clearly U and V are disjoint neighbor-hoods of p and q respectively. Case (3) : G is a subspace of o<X. There exist disjoint open subsets U and V of G containing p and q respectively. G i s open in o<X, so U and V are open in p<X. Since h(U) = U and h(V) = V, 4 4 U and V.are open in Y X . Therefore TX i s a compactification of X . 1 - 2 . 4 Lemma : Let X he locally compact and let «XX be a compactification of X with (3-family J(cAX). Suppose K1 , K 2 6 ? ( * X ) , let J* =(? (*X) - 4 K 1 , K 2})U [^U K2\ . Then there exists a unique compactification TX of X with J ( r x ) = J* . Proof : -Let f^ be the p -function of <X.X, q^ be the point in dX- X such that K ± = f j 1 (q i) i=1,2 Let YX = («<X - ( q ^ q 2 } ) U { A} where A is any point not in o(X. Let h that maps o(X to Y X be defined by h(p) = p for a l l p € c<X - {q 1 ,q2}, h-Cq^ ) = A i = 1 , 2 and let YX have the quotient topology induced by h. Then Y X is a compactification of X ( by Lemma 1 - 2 . 3 ) . The function h'f^ mapping (3X to Y X i s a continuous mapping from j3X onto Y X such that h'f^ (x)=h(x)=x for a l l x e X . Since the p-function of a compacti= fication i s unique, h 0 ^ must be the p-function of Y X , so 7 ( Y X ) = J*. The compactification Y X i s unique since any comp-actification is uniquely determined by i t s p-family. 45 1-2.5 Lemma : Let X be locally compact, and {E^: i=1,....,n} be a fini t e family of mutually disjoint, 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 is a special case of Lemma 1-2.3 with c*X = |5X. 1-2.6 Definition: The unique compactification TX of Lemma.1-2.5 w i l l be denoted by p((X; , K n) throughout this section. In case that there exists a compacti— fication SX of X such that J ( SX) consists of an infinite number of closed sets K^ of p x together with a l l singletons {p} where p e(|3X - X)- ^ ^K^ > we denote 5 x by o<(X; , Kg,....). 1-2.7 Lemma : Let X be locally compact, and let K^  and Kg be two non-empty closed subsets of pX - X. Then (1) ^(X;^) A p((X;Kg) = d (X;^ , Kg) i f K^  C\ Kg=0 (2) odliK^AodXjKg) = oUXj^UKg) i f K^  fi Kg^ 0 . 46 Proof :-(1) If K 1H K 2 = 0 , then by Lemma 1-2.1, we have P U X J K ^ ) i eUX;^ ,K2) and *<(X;K2)=s O<IX,K 1 ,K 2). Let YX be any compactification of X such that P U X ; ^ ) ^ YX, *U;K 2) ^  YX. By Lemma 1-2.1, i s a subset of some set in J ( Y X ) and i s a subset of some set in 2( YX). So, o((X;K1 , K 2) * YX; and c^CXjK^ A ^(X;K2) =C^(X;K1,K2). (2) If E ^ H K 2 £ jZ), then by Lemma 1-2.1, o^CX;^) ^  oCU^UE^), o((X;K2) ^  * ( x j ^ U K 2 ) . Let YX be any other compactification of X such that ot(X;X1 ) ^  TX and o^(X;K2) ^  YX. Then i s a subset of some set in •?{ YX) and K 2 is a subset of some set H 2 in -F{ VX). Since 3Fi, YX) is a partition of |3X - X, and fl K 2 j£ 0, we have H1 = Hg. Thus K1 V KgSHj and ^ ( X j K ^ K 2) YX. So rifXjK,,) A *(X;K2) = (X (X;K1 U K g). 47 1-2.8 Lemma : Let X be locally compact and let o<X be any compactif ication of X. Suppose , IL, g J(<*X) and that , are non-empty closed subsets of H1 , H 2 respectively. Then the (3-family of o<X A o((X;K1 U K 2) is (J( o(X) - { H1 ,K2) )u { H1 U Hg} Proof :-By Lemma 1-2.4, there exists YX, a unique compactification of X such that 3( YX) = (J(oCX) - {H rH 2D U f ^ U H2}. (a) o(X = YX since every set in J( oCX) is a subset of J( YX). (b) ^(X;^ U Eg) ^  YX since K1U K 2 £ ^ U Hg and so every set in J( ^ (XjK^UK^)) i s a subset of J( YX). (c) Suppose SX is a compactif ication of X such that e<X = SX and ^(XjK^UK^) = £X, then every set in ^ ( o(X) i s a subset of some set in 3 { SX) and every set of J ( o((X;K^ UKg)) i s a subset of a set in ?( SX). Therefore H1, H2, K 2 are subsets of some sets of J ( 8 X ) . let £ j U K ? c 5 €J(SX), R\C T e J( SX) and H ^ U e J( SX). Since K^c ^  , K 2c H 2 and 5( SX) is a disjoint family of sets, we have T=U and so H ^ H ^ T € 7( SX). Thus YX = SX, and therefore 3( « A ^(X^UKg)) = J(<*X) - ^  H1 , H ^ U ^ l i H ^ . 4 8 1-2.9 Definition : A compactification dX of a locally compact space X i s called a dual point of the lattice K(X) i f dX ^ px and there exists no compactification YX different from both dX and |3X satisfying dX < YX < pX. 1-2.10 Theorem: Let X be locally compact. Then dX. is a dual point of K(X) i f and only i f there exist distinct elements >p and q of )5X - X such that dX = o^(X; {p,q} ) Proof: -(1) Necessity : If dX i s a dual point of K ( X ) , then dX <t px and there does not exist any TX; |3X ^ YX £ dX such that dX < YX < px. Since dX < {JX, . every set in J{ (3X) is a subset of 3(dX). Also, <*X £ px. (a) There exists one and only one set K in 5 ( dX) that contains more than one element of 0X. ( i ) there exists at least one such set, otherwise dX = |3X. (ii) If there exist two or more such sets namely: ,.... ,K n 2 , then o K X ; ! ^ , , K n ) =o(X and /?X > ^ (X; !^ )>*<X 49 This contradicts the definition of <*X as a dual point of K(X). So (XX = <X(X;K) for 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 distinct elements of |3X - X. Since K contains more than one element of /3 X - X, let K 3 {p,q}for some distinct p,q e )3X - X. Suppose r i s distinct from p,q such that r € |8X - X and re K. Then ={p,q>is a set that satisfies the following inequality: <XX = <rf(X;K) £ ^(X;^) £ /JX This contradicts the definition of <XX. (2) Sufficiency: Let oCX = <X(X;{p,q}) for distinct p,q € pX~ X. Clearly, o(X < (3X. If there exists YX such that dX < YX < j3X and ($X ^  YX ^ c<X then there exists F e J ( YX) such that F c {p,q} and F ?Mp,q} since P^X £ YX. Then F =(p) or F = {q} and this implies that YX = px. It i s a contradiction. 50 1-2.11 Theorem : Let tfX be a compactif ication of a locally-compact space X. Then J( *<X) has exactly one set which is not a singleton i f and only i f (a) °<X ^  J3X and (b) there do not exist two distinct compacti-fications TX and SX such that (/') both TX and SX are dual points. (ii) o(X A YX = c*X A SX ± <kX and (iii) 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 is 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 in /3X - X such that YX = <X(X; {a,b} ) and %X = << (X; {c,d} ). Since <*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 esse n t i a l l y three dif f e r e n t cases to consider : (1) a, b, c, d t K (2) a, c i K; b,d eK; (3) a, b, c 4 K; d e l . In the f i r s t case : <*X = oC(X;K) and 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} ) = o<(X;K, {c,d} ) so{a,b}= {c,d} which i s a contra-d i c t i o n since YX = d{X; {a,b} ) and $X = d ^ X; {c,d} ) are d i s t i n c t . 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) and since a ^ K , c ^ K, a = c . So, b ^  d because YX and SX are d i s t i n c t . Again, by Lemma 1-2.7, <rf(X; fa,b) ) A <<(X; {c,d} )=tf(X; {a,b,d} ). C l e a r l y , o((X; fb,d}) i s a d u a l p o i n t g r e a t e r t h a n tf(X;{a,b,d}) and d i f f e r e n t from b o t h o<(X;{a,b} ) and <*(X; {c,d}). So (ii!) i s not s a t i s f i e d . I n the t h i r d case : Lemma 1-2.7 i m p l i e s t h a t <*X A o((X; {a,b} ) = c*(X;K,{a,b) ) and dX A e<(X; {c,d}) = <* (X; K U {c) ). C l e a r l y o((X;K, {a,b} ) ^ ( X ; K U ( c l ) . Thus, c o n d i t i o n ( i i ) i s not s a t i s f i e d . So we conclude t h a t t h e r e do not e x i s t two c o m p a c t i -f i c a t i o n s YX and SX which s a t i s f y c o n d i t i o n (/), ( i i ) and ( i i i ) . (2) S u f f i c i e n c y : From ( a ) , <<X £ pX, t h e r e e x i s t s a t l e a s t one s e t K i n J ( <*X) which c o n s i s t s o f more t h a n one element. Suppose ? ( o<.X) has a n o t h e r such s e t H. 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. Then {a) = H, (b) H, { c ] c K, {dj c K. L e t ( i ) YX = PC(X; {a,c>) and SX = <*(X; {b,d}) . 53 By Lemma 1-2.8, the 0-families of 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 uniquely determined by i t s P-family; and cl 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 dual points of 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 greater than << (X; {a,c}, {b,d} ). Then either {x,y}c{a,c} or {x,y}c{b,d}. Equivalently, either o((X;{x,y} ) = YX or U{X; {x,y} ) = %X. Thus we have (m) the only dual points greater than YX A SX are YX and SX. Now we have shown that YX and 5X are compactifications of X s a t i s f y i n g (/), (/'/') and (/'/;) of (b) above; which i s a contradiction. So J(c(X) has exactly one set consisting of more than one element. 54 1-2.12 Lemma : Let X be locally compact, <*X a compactificat-ion of X with )5-family J( d X) and H a closed subset of /3X - X containing more than one point. Then H e J ( * X ) i f and only i f <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 : If YLeS{dX)y then (a) P<(X;H) ^ *X since every set of J(oC(X;H)) is either a singleton subset of /3X - X or H, so i t i s a subset of some set inXX. (b) If 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) is a decomposition of (9X-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 oC(X;K) such that rt(X;H) > d(X;X) * dX. so H = T, . H € J( oCX). 55 1-2.13 .Theorem : Suppose that X and Y are locally compact and that T is a lattice 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. If card ( |3X - X ) £ 2, the homeomorphism is 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 flX, the one point compactification of X . Hence K(Y) consists of two elements and i t follows that pY - Y consists of two elements. There is only one isomorphism from K ( X ) onto K(Y). However there are two homeomorphisms from px - X onto pY - Y, and "both satisfy the condition of the theorem. Suppose that px - X has three or more elements: (1) 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 in /?X - X such that p,q,r are distinct. 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 is 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 is 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 flX - X and s i s different from p , q and r. Then there exist y and a in p1 Y - Y such that F(oC(X;{p,s) )) = ^ ( Y ; f y , z i ). We have T( (X; {p,q} )) = d (Y; {a,bl ) and we may assume that rU(X; {p,r} )) = oC(Y; {a,c} ). Using the argument given above, we have-fy,z} intersects both {a,b} and {a,c} in one point. Thus i f a 4 fy,z} , 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 for any S€(3X - X - ^  i f I U U ; { p , s } ) ) = * l Y ; f y , z > ) then a e {y,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 distinct points of H, and T( <X (X; {p,q} )) = *(Y; {a,b} ). Then by Lemma 1-2.2, <x(X;(p,q}) ± U{X;R). And then f ( *(X; fp,q})) = f(<X(X;H)) that is o((Y;{a,b}) a s c ( ( 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 0Y - Y into pX - X in the same way as h : Given a e pY - Y, choose distinct b,c e |8Y - Y. This is possible as K(X), K(Y) are isomorphic and px - 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~ 1 (o((Y; {a,b})) and r " 1 ( o( (Y; {a,c})) are dual points of K(X). Therefore there exist points p,q,r,s in /SX - X such that IT 1 (o((Y; {a,b})) = o((X;(p,q}) 59 and r " 1 (eA ( Y ; {a,ci)) = d ( X ; {r,s>). {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 for h, we have ( m ) k(K)cH for the sets H and K of ( i ) . Now let p,q be distinct points in px - X , then 37( oi ( X ; {p,q})) = o((Y;{a,b)) for some distinct a,b in 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 distinct from p and q, then there exists c « p Y - 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 is 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 P Y - 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 i s h . Therefore h i s a homeomorphism. 60 Suppose TX is a compactification of X with J5-family 3( Y X ) , and T( YX) = TY. Then ?( YY) = fh.LHJ : H € 3( Y X ) } : Let H 6 J( TX) and H contain,, more than one point. Then oC(X;H) = YX . By (iv) above, c<(Y;h(H))^ YY. By Lemma 1-2.12, there exists no compactification of the form <*(X;V) such that o((X;H) > *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). Similarly, i f K e J(YY) and K consists of more than one element, then h"1(K) £ J( Y X ) . Now consider a singleton /p} in J ( YX) i f i t exists. If fh(p)} i s not in 3( Y Y ) since h(p) e 0Y - Y and 3 (YY) is a decomposition of PY- Y, there exists a subset K containing more than one point in 3 (Y Y) such that h(p)e K. —1 —1 Then p £ h UL1 where h IK1 contains more than one element. This is a contradiction since h"1 CK3 e J ( Y X ) , and the sets in 5"(YX), are mutually disjoint. Similary, if{a) e J(YY), then {h~ 1 ( a ) h 3 { Y X ) and so ^(YY) = {hCHI : H e J ( Y X ) } . 61 (4) The homeomorphism h i s unique: Let t be a one-to-one mapping of j3X - X onto j8Y - Y such that for any compactification YX of X , i f r ( T X ) = YY, then J ( Y Y ) = f t £ H ) : H e J ( T X ) } . Given p e px - X and choose two other points q, and r in px - X such that p,q,r are distinct. Then there exist distinct a,b,c in (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 b ^ c as T is one-to-one. At the same time, tlfp,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 is a contra-diction. 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 locally compact and that h is a homeomorphism from j8X - X onto j8Y - Y. Let dX be a compactification of X with j?-family 3(oiX). Then there exists a unique compactificat-ion o<Y of Y whose /3-family is {hOD : E e J{ P<X)}, and the mapping 7 defined by I( i/,X) =«<Y is a lattice 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 function mapping / J Y - Y onto o<X - X. Let <KY = Y U [ o(X - XI, and define a — 1 function k mapping JJY onto °(Y by k(p) = f^ • h~ (p) for p e f3Y - Y :, k(p) = p for p i n Y . Let o(Y have the quotient topology induced by k. (1) o(Y i s compact with 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 compact. For any open set Og i n <*Y, there exists an open set 0^  i n 0Y such that klO^JCOg. Since Y i s dense i n ^ Y , t h e r e exists p e Y 0 0^  such that k(p) = p € 0^  . So Y i s dense i n o<Y. It i s clear that Y i s an open subspace of oCY since Y i s open dense i n 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 points p and q i n o(Y we have three di 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 i n j J Y , k~ 1(q) i s closed i n /SX and, so i s compact. There exist open subsets U and V of j3X such that p e TJ cY, k"1 (q) e V, and U A V = 0 Let U*= kCU] , V* = k m . 63 It follows that U and 7 are disjoint open subsets of d~L containing p and q, respectively. (ii) Both p and q belong to Y : Y i s open in |3Y, there exist open subsets UcY, V c l 0 f j3Y such that UD V = 0 and peU, q e V. Since Y i s open in c<Y and U = k~1CUT, V = k"1.CVl U and V are open in 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 P q. containing p and q respectively such that C1G f l C1G = 0. Then k" 1(ClG ) and k~^(C1G ) are disjoint closed and hence compact subsets of 0Y - Y. Then k" 1(ClG ) and k~1(ClG: ) are disjoint closed subsets of /3Y which is compact, normal. There exist disjoint open subsets H and H of J * p q fJY such that k~ 1(ClG p)£H p and k" 1(ClG )SH . Since k~1 (G ) and Pi. PJ. P _ A k (G ) are open subsets of j3Y - Y, there exist two open subsets U and 64 U of (3Y such that k~1 (G ) = U n (jSY-Y), q r P P * ' k~ 1 (G ) = U fi ( BY - Y). q q K Then we have k~ 1 (G ) = H f) U n ( B Y - Y) P P P K k~ 1(G ) = H ft U n (pY - Y). Now let, V = (H A U 0 Y) U G and P P P P V = (H H U 0 Y) U G . Then V , V q q q q p q are disjoint subsets of e<Y, containing p and q respectively. k"1 (V p) = k"1 (H p0 U pAY)U k"1 (Gp) = ( H p n u p n Y ) U (H pn u p n ( R Y - Y ) ) = (H n u ) H IY U (BY - Y)] p p V K p p which i s an open subset of (5Y. Since oCY has the quotient topology induced by k; V is open in <*Y. Similarly, is open in e<Y. Prom (/), (;j) and (iJi) we have o<Y a Hausdoff compactificat-ion of 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 in the p-family J(o<Y) of «<Y. Then there exists a point p i n oCY-Y=o(X-X such that K = k" 1(p) = l f ^ - h " 1 ) " 1 C p ) = h(f* 1(p)). Thus K i s the image under h of a set in J(<*X), 65 the p-family of (*X. On the other hand, i f H is any set in ^( r f X ) , there exists a point q in o<X - X such that H = f^~ 1(q). Then h(H) = h(f,~ 1(q)) = U r f - h ' W ) = k' 1(q) in 2(cXY). 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. Define T{ d X) = <XY, then (4) T is a lattice isomorphism from K(X) onto K(Y): If YX and SX are two different compacti-fications of X, then the two p-families J( YX) and .?( SX) are different. 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 is any compactification of Y with p-family 3-( YY), then there exists a unique compactification YX of X such that J ( Y X) ={h"1(H): H e J ( Y X).} Thus r( YX) = YY, and i t follows that T i s a bisection from K(X) onto K(Y). The fact that P is a lattice isomorphism is a consequence of the fact that the following statements are successively equivalent : YX ^ SX; each set in J(SX) is a subset of a set in J ( YX); each set in {h(H) : H e J ( % X)} bb is a subset of a set in {h(H) : H € 3= ( YX)} ; T( YX) ^ T( S X). Thus the proof of the theorem is complete. 1-2.15 Corollary : Suppose that X and Y are locally compact. Then the lattices K(X) and K(Y) are isomorphic i f and only i f |8X -X and ]3Y - Y are homeomorphic. CHAPTER II HOMEOMORPHISMS OF p N In this chapter, we study homeomorphisms onto (3N. 2-1.1 Definition: If fl^ , &2 a r e -ultrafilters on countable sets X 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 corres-pondence f of X onto Y such that for each Ec:X, Ee i f and only i f fCE] eQ 2; or equivalently, pe |3X, qe (3Y are of the same type i f EcX, p e C l ^ E i f and only i f q C I D V f CE]. p i 2-1.2 Definition: If K is a permutation of N, fl^ is an u l t r a f i l t e r on N, fig = K( ) is the u l t r a f i l t e r which contains the sets Jl(E) for a l l E e ff^ . 2-1.3 Remark: It follows that i f two ultrafilters D.^ , ft2 on N are of the same type, then there is a permutation K of N such that n 2 = 71(1^). There are 2 C types of ul 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 ) If K i s a permutation of N, then it i s a con-tinuous function from N into pN. Thus ft , the extension of K to pN is a continuous function from pN into pN (By Theorem 0 - 2 . 4 ( 1 ) ) . K* i s onto since ix t (JN.1; is compact and N i s dense in pN. For distinct x,y in pN, there exist disjoint A, B subsets of N such that x € Ol^jjA , y e Ol^B because pN is Hausdorff and i ^ l ^ A:AcN) forms a base for pN. (Remark 0 - 3 . 2 ) 7C (x) € C l p * CAI and 7t*(y) £ Clp N 31 CBI . Since * EA3 Hit LB] = 0 and nlAl, n CB] are zero-sets of N, so 0 1 N1T £A1 r i C l B N ft.IB] = 0. Thus 7t (x), K (y) are distinct. Therefore 7T is a homeomorphism of pN, and i t is 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 precisely c homeomorph-isms. 2-1.5 Theorem: Por free u l t r a f i l t e r s , on N, there i s a homeomorphism of (3N which carries "to O.^ i f and only i f fl^ and a r e isomorphic as p a r t i a l l y ordered sets. Proof:-As a result of Theorem 2-1.4, we need to prove that there i s a permutation n of N such that = J t ( ) i f and only i f Q.^ and Cl^ are isomorphic as p a r t i a l l y ordered sets. (1) Necessity i s obvious. (2) Sufficiency: I f and a r e isomorphic as p a r t i a l l y ordered sets, there i s a one-to-one mapping f of onto SXp such that A c B i f and only i f f'EAD c f l B l , and c l e a r l y fCNI = N. l e t H n = N - {n} , c l e a r l y the H n are maximal proper subsets of N and are permuted by f. Now define a permutation n of N such that TT (n) = m i f ±'(H ) = H . For any E « i , l e t n m • i F = N - E, then E = H H . Let ' neF n E ' = n?F f ( H n ) . Since E c H n f o p ^ n e F > F ( E ) c f ( H n ) for a l l n e F ; so F ( E ) c E ' . 71 S i m i l a r l y , E'c:f(H^) for a l l n e F , f " 1 (E' ) c H n for a l l neF; so f " 1 ( E ' ) c E and therefore f(E) = E*. But >• - n % f(H n) = nF H „ ( n ) = n ? (B- „<„)) - N - nyF^ („) = N - JTI (n): n e F} , so N - E* = {n (n): n e F} that i s N - f(E) = K I N - E ] so f(E) = 71 (E). The theorem follows. 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 of N such that X ( Ii 1 ) = £i 1 and 71 ( _p_2) ^ i l 2 Proof :-i l ^ ^ Hg, there exists E^c N with i n f i n i t e complement such that E 1 £ ^ and N - E^ € (By Theorem 0-1.6) Choose £Lj £ i l 2 such that N - e 12^. Sim-i l a r l y there exist i n f i n i t e sets Eg and E^ such that Eg e Ug, E^ e Si^, EgU E^ = N - E t,and E 2 n E^ = 0. Let 7[ be a permutation of N such that X (n) = n for n e E^, n(n) i f n e Eg and 7 t(n)eEg i f neE^. Clearly n( si^) = and u ( lig) ^  li-g. 72 2-1.7 Corollary : If a, b are distinct elements of (3N then there exists a homeomorphism 7T of (3N such that 7f (a) = a and K (b) £ b. 2-1.8 Theorem : If A' , B' are non-empty open-closed proper subsets of N , then there is a homeomorphism of £N which maps A' onto B*. Proof :-A', B1 are non-empty, there are infinite A, B * subsets of N such that c l ^ A fiN = A' , c l ^ B f i N * = B' (Theorem 0-4.1) A' ^  N* £ Bf implies that N* - A' and N* - B' are non-empty open-closed subsets of N . Thus N - A, N - B are infinite subsets of N. Let 7i be a permutation of N which maps A onto B and 71 induuces a homeomorphism of /3N. CHAPTER III 73 HOMEOMORPHISMS OF N In this chapter we study the homeomorphisms of * N . Most of the results of t h i s chapter depend on those of chapter I I . The continuum hypothesis i s assumed through out t h i s chapter. 74 3-1.1 Theorem : * ,c N has 2 P-points, and the set of P-points of N is dense in N . Proof (1) There exists P-points in N . Let {WJ} , the family of a l l the open-closed subsets of N he well- ordered with Wj = N , where <*. runs through the countable ordinals. Now select {A^} a family of open-closed subsets of N in the following way : A» = N For o< , a countable ordinal such that an open-closed set A£ has been selected for each p < <>< so that fl { A£ : 0 < * } = B* is not empty, by theorem 0-4.2 there exists a non-empty open-closed set G^' such that Q* C B 1. If C; n = 0, set A J = Ci . If CJ 0 WJ £ 0, set A,J = G' 0 WJ The family { A J 1 : runs through the countable ordinal} so formed is linearly 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 the compact space N w i t h the f i n i t e i n t e r s e c t i o n p r o p e r t y . (b) A' = ( a 1 } . I f a f £ b» € A', t h e n a' 6 ¥' , b* e ¥' f o r some r,p o f r P the c o u n t a b l e 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} forms a b a s i s o f N ; and the c h o i c e o f AJ shows t h a t i f A'H ¥| £ 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 i o n . c l a i m t h a t a 1 i s a P - p o i n t : (a) The f a m i l y {Aj} where ai runs t h r o u g h the c o u n t a b l e o r d i n a l s , forms a b a s i s a t a 1 s i n c e f o r any open neighborhood CT o f a', t h e r e e x i s t s a c o u n t a b l e o r d i n a l K such t h a t a'e W^CG, so a 1 6 AJcr¥J c 6 . (b) I f [ G ± : i e N } i s a c o u n t a b l e c o l l e c t -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 e x i s t s <*± such t h a t A ' . c e . , I f <* i s the s m a l l e s t o r d i n a l w h i c h exceeds ev e r y o t i , i e l f t hen A^ cr ^ ^ i * * So a* i s a P - p o i n t o f N . 76 c * There are 2 P-points i n N . This i s shown by the fact that i n the preceding construction, there are at least 2 d i s j o i n t candidates for each i n each of the c stages. In Theorem 0-4.2, the construction of A shows that we can have B with the same property as A, and BfiA = 0 by choosing k m^  e A^  f\ A 2 0 .... f\A k - jiU1 n^ and n k + 1 € A i n A 2 n " - - ° A k n A k - M " h^1 "h (this i s possible because A^  fl A 2 0 ... 0 A^ . H A^^ i s i n f i n i t e ) . Then A, the set of a l l n^ chosen and B, the set of a l l m^  chosen are d i s j o i n t sets with the required property. The set of P-points of N i s dense i n N This follows from Theorem 2-1.8 that for any non-empty A', B* proper open-closed subsets * of N , there i s a homeomorphism of 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 is not homogeneous Proof :-If every point of N is a P-point, then the inter-section of any countable family of open sets is open; so the union of any countable family of closed sets is closed. Then every countable subset of N is closed and discrete ( by Theorem 0-1.16(2) ). This * is impossible because N ±s infinite compact. N is 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 f then there is a homeomorphism of N 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 open-closed subsets of N that contain a' such that Aj = N* and let Similarly, { B'} is for b 1, B' = N* and V = - BJ . Now construct a permutation ip of the family of a l l 78 open-closed subsets of N such that (a) lp (A' )c<p (B f) i f and only i f A' cB» (b) (pmaps {A^ } onto [Bj] . Let 9(Aj) = Bj , <P(X') = Yj and proceed by trans-f i n i t e induction as follows : suppose c< i s the smallest ordinal f o r which <P (AJ< ) has not yet been defined such that the sets for which 4> has been defined form an at most countable rin g (R.| , and ip preserves f i n i t e unions, f i n i t e intersections, complementations, and inclusions. Since a', b' are P-points, there are sets AJ and BJ ( f > <*) such that AJ cA^ with A} i n the intersection of a l l A£ f o r which ip has been defined, and BJ i n the intersection of the corresponding sets <P (AJ ) . Define if{A\ ) = BJ and <f(XJ. ) = YJ Now the smallest ring <R2 that contains (R^  , AJ and XJ consists of a l l sets of the form ROAJ , ROXJ (Red^) and f i n i t e unions of tnese sets. Either RflAJ = AJ. or RflAJ = -0 ; so we need to define cp on {R n x ; : R€(R.1 } , {(R1 0 X' ) U R 2 : , R 2 € } and { AJ UR : R « ^ l . Let q>U nxj ) = ip(R) n Y; I?( R U A J ) = ip(R)UBJ and ( piR^XJURg) = lf> ( R ^ O Y J ^<P(R2) . 79 The function if i s now defined on a countable r i n g (R-2 a n ( i y preserves f i n i t e unions, f i n i t e i n t e r -sections, complementations and inclusions. Now, divide the members of (ft2 into three classes {F-}, ( G J , {E^} such that A ^ F , , G j C A * a n d n 0 inclusion holds between and for a l l ke N. Put S ± = F ^ P g O HF i T . = G U G„ V U G . then we have and H k * T j ' f o r a 1 1 k , j e N , for T c Afc and i f H k c T. then H k c A^. Similarly, H k for a l l i.k e N ; tf(T^ V ( T n ) f p y ( S n ) f ... ^ ( S ^ and 9 ( 1 ^ ) ^ ^ ( 1 ) WS^ VU^ ) for a l l i, 3 , k € N. By Theorems 0-4.4, 0-4.5 there exists an open-closed set Z i n N different from any set so f a r i n the range of f such that ( f ( T . ) c Z C ^ ( S . ) for a l l i , j , € N, neither of the sets <f (H k) and Z contains the other and (f(Q)DZ £ 0 whenever QHA* ^ 0. Since AY<= A*, B,cZ and Z i s a member of the family {B„} . Define ( fUj = Z and ip(X„) = N* - Z . Let (R^  be the ring generated by |R,2 and A^ : consists of a l l sets of the form ROA* , R n X A (Re <ft_) and f i n i t e unions of these sets. Define q>(RO = </>(R) n cP(A*) , f l R H X j = < P ( R ) 0 «?(X^) , and <p(PiVQ) = ¥(P)U<P(Q) i f <P(P) and cp(Q) are defined. In t h i s way tf> i s extended to the ri n g dl^ so as to preserve f i n i t e unions, f i n i t e i n t e r -sections, complementations and inclusions. Clearly a'e R i f and only i f V e <f(R) for a l l Re (R^  . Similarly cf~1 i s defined to s a t i s f y the same induction hypothesis for 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 ri 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 , for any a e N there i s one and only one h(a) € N which i s contained i n ¥>(A') f o r every open-closed set A ' which contains a : (a) if i s defined 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 also open-closed i n N ; either a'e X or a'f N - X. . Hence <f i s defined on X; and so i s <p~^ . So (/> permutates the family of a l l # open-closed sets of N . (b) For any a e N , l e t i A r f) be the family of a l l open-closed subsets of N that contains a. We claim that P\ {k^) = {a\ : Por any a^b e N , there exist open-closed A, B such that a e A, b e B and A.HB = 0 ; that means there exists A e {A*} such that b A; so b 4 f){Aj . (c) f] jlfCA )^} i s non-empty. {<f(A^)\ i s a family of open-closed subsets of N which i s compact. Since there exists B, an open-closed set such that B e n (A^.} f o r {A^ .} , any f i n i t e sub-family of {A^} ; so <f(B)c f l j^U^)} that i s {lf(A^)) has the f i n i t e intersection property. Thus f l l ^ t A ^ ) } i s non-empty. (d) Let b 6 fl{cf ( A j } , then ft {<f (A^)} = {b) . For any b p c € N , there i s an open-closed C such that c € C and b 4 C. Thus C 4 {^(A^)} since b € HfyU^)}. Now (f"1(C) and if"1 (N* - C) are complementary. So either if~ 1(C) or vf " 1 (N* - C) belongs to (A^)and i f " 1 (C) does not, therefore i f " (N - C) £{A*}. There i s D = l f " 1 ( N * - C)e { A j s u c h that c 4 v(D) So c 4 n f v u j } (e) Define h(a) = b . Since If maps {A^} onto |B^ } , h(a*) = b' and i t i s clear that h so defined i s a homeomorphism. The theorem i s proved. 82 3-1.4 Theorem : * c N has 2 homeomorphisms. Proof :-U) There are 2° p-points ( Theorem 3-1.1), by-theorem 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 distinct homeomorphisms induce distinct: permutations. * c N has at most 2 homeomorphisms. 3-1.5 lemma : If X i s a countable discrete subset of N , then for each a e X there exists E' so that a a e E' and { E 1 : a e X } is a family of disjoint sets where E c u and { E : a e X } is also a family of disjoint sets. Proof :-Since X is a countable subset of N , let X = | a,j, &2 > ••••}• Now let us construct E^ as follows : (a) E 1 : {a^} is open in X, so there i s c N such that a 1 € F ' and Fj f) X = {a^. Set E 1 = F 1 . 8 3 (b) E 2 : | a 2 } i s open i n X, so there i s ~F^a N such that a 2 e F 2 and F£ fi X = { a 2 | . Since a 2£ implies that a 2 f G l p N E 1 ^ ^  £ 0 1 ( N _ ^ ) , A N D a 2 € C V V o r a 2 e C 1pN ( F 2 n N " E 1 ) Theorem 0-2.4(4). Set E 2 = F 2 0 ( N - E 1 ) . (c) E ^ while E i s defined whenever j< i : { a^ } i s open i n X, so there i s F^c N such that a. € F! and F! fl X = f a.}. Since a - 4 E'. whenever J < i , a. € C l ^ (N - .U l E .) and a. * 01 ^  F.. Thus, a. € 01 R „ (F. fl (N - V)1 E i ^ - S e t  1 PN i i i jWl J E . = F i n ( N - d u 1 E . ) . The family { E ^ : i € N } constructed s a t i s f i e s the conditions required. 3-1.6 Lemma : If XUY i s a countable discrete subset of N and a' i s i n the closure of both X and T, then a' i s i n the closure of XOY. Proof :-Suppose on the contrary that there i s an E « = K so that a'e E 1 , and E ' f ! (X fl Y) = 0 where E' = CI N EHN*. XUY i s countable and discrete, so by Lemma 3-1.5, 84 for each b e X UY there exists EJ such that b e EJ b b and {E£ : b € X UY} i s a family of d i s j o i n t sets where E^C N and {Efe :beXUY) i s a family of d i s j o i n t sets also. Let E x = {n e N : n e (E bn E) for some b e (X f l E 1 )} E = {n;e N : n e ( E b 0 E) for some be (Y HE')} . Since a 1 i s a l i m i t point of X and a'e E 1 , a' i s a l i m i t point of X HE' (that i s a' i s i n the closure of X / 1 E ' ) . E' i s an open-closed set containing -X. X H E 1 . Hence, C l a w ( X D E•)D N*C E* and so a'eE'. pi* X X Similarly, a ' e E 1 . But E A E = 0 : suppose y;' x y p e E HE , then by d e f i n i t i o n , x y p e ( E b H E ) for some b e (XH E') and p e (E n E) for some a e (Y H E 1 ) . So, p e (E n E. ) . But E , E,are d i s j o i n t . So E'HE' =0 ; that i s a contradiction. x y 3-1.7 Theorem : If X i s a countable i n f i n i t e discrete subset of N * f then C l p N X = p X , C 1 ^ N X i s homeomorphic to p N . Proof :-Let 71 be any one-to-one correspondence from N onto X . 7C i s a continuous function from N into C l p j j X. Then 7[ the extension of 7T to p N i s (a) continuous from p N into 01^ X; (b) from p N onto Cl^ j j X, since 7t ipN3 i s closed; (c) one-to-one : since N i s Hausdorff. I f x, y are d i s t i n c t elements of GN such that K : (x) = TT (y) , then for any neighborhood U of 7T*(x) i n C1^ N X, there are N(x), N(y') neighborhoods of x and y i n p N such that N(x)0 N(y) = 0 , JT*£N(x ) ]cu and 7[*[N(y)l C U . Then we have 7t*(x) € 0 1 ^ 7C*[N(x) n N3 and / ( y ) e CI N * * I N(y) AN] . Now, 7T CN(x)ON] U3i 'CN(y) ftND i s a countable * discrete subset of N ; C 1 p N J I * I N ( x ) A N 3 = C 1 N # 7 T*£N(x ) nN ] and C l p N 7T*CN(y) AN] = C1 N^ 7 i * C N ( y ) n N 3 . By Lemma 3-1.6, -K*(x) € CI N {it* CN(x) fi NI H 7C* CN(y) AN]} which i s empty since 7C IN i s one-to-one and N(x) A N(y) = 0. (d) closed. So G l p ^ ^  i s homeomorphic to pN. Since ji i s a homeomorphism of N onto X, both of N and X are discrete countable, then 7 1 ' , the extension of 71 from pN to |JX i s a homeomorphism. So G l p j j X i s homeomorphic to |3X; or equivalently, CXpjjj. X = px (by the uniqueness of px f o r given X). 86 3 -1.8 Theorem : If for each ncN, h n i s a homeomorphism of N onto N and (IT } and {^(tM)} a r e countable •it-families of d i s j o i n t open-closed subsets of N , then there i s a homeomorphism h of N onto N such that for each n e N and xeU', h(x) = h (x). n' v n Proof :-Let x^ e Uj be a P-point, y 1 = h 1(x^) ; y 1 € h 1 (U|). Let fi.^ be the ring generated by IT U {tP} We define ip a mapping of ft^ into the family of a l l open-closed sets of N i n the following way: ip(U£) = h ^ U p for a l l i € N , 9(0) = 0 , V(N*) = N*» if(N - P) = N - ip(P) whenever cp(P) i s defined, f ( p UQ) = ¥ ( P ) u <P(Q) » f(PAQ) = <p(P )0(p(Q) whenever «p(P) and i p ( Q ) are defined. Now ip i s defined on (R^ such that (/> preserves f i n i t e unions, f i n i t e intersections, complementations, and inclusions. Let {W^ }. be the family of a l l open-closed sets containing x^ , iU^} be the family of a l l open-closed sets containing y^ . Arrange (WJ} - ( R j and {u j } - { i p ( R ) : fie^] i n order as {A;} and {B^ } respectively, where <?< runs through the countable 87 ordinals. We extend (f to the ring generated by {A(j}u(R1 onto the ring generated by {BJ}u{ip(R) : Re R^} in the following way : Suppose <<^  i s the smallest ordinal for which <P(AJ[) has not yet been defined such that the sets for which if has been defined form an at most countable r i n g ^ and if> preserves f i n i t e unions, fi n i t e intersections, complementations and inclusions. Define Qn = AJ^  f\ U n for each positive integer n; if>CUn - Qn] = h nCU n - QN3 , and extend ip to the ring <ft5 generated by -CUn - Qn> U -CQ^  UR 2« Then extend <P to the ring generated by {AJ^ } u (R^  by the same method in Theorem 3 - 1 . 3 . Then the theorem follows. 88 3-1.9 Theorem : Suppose X and Y are countable sets of P-points of N and p and q are l i m i t points of X and Y respectively. There i s a homeomorphism of N onto N which carries p to q i f and only i f p and q are of the same type. (Definition 2-1.1) Proof :-(1) Necessity : X and Y are discrete (Theorem •jt-0-1.16(2)). Let h be the homeomorphism of N onto N such that h(p) = q . Select subsets X^  and Y^  of X and Y respectively so that X - X^, Y - Y^, are i n f i n i t e and p € C l ^ X ^ , qe 0 1 ^ N Y^. Let Y 2 = hCX^ H Y1 and X g= h" 1 CY^ . By Lemma 3-1.5, q e C 1 ^ N Y^  and 01 p N h l X ^ ^ h C X ^ ^-Cql, then qe 01 ^ Y ? , and pc 01^^ X 2 . Clearly h induces a one-to-one correspondence between X 2 and Y 2 such that for each E<=X2 , and p e C l ^ E implies that h ( E ) c Y 2 , so q e C l p N h(E) . Let q be any one-to-one corres-pondence from X -X 2 onto Y - 1^. Define f as the one-to-one correspondence from X onto Y such that f(x) = h(x) for x e X 2 and f(x) = g(x) for x e X - X 2 If E<=X, then p e C l ^ E i f and only i f 89 p i s a limit point of E D X^ ; hence p e C l ^ E i f and only i f q is a limit point of f(E) fiY^. Therefore p and q are of the same type since i f EcX, p € Clp N E i f and only i f y e 01^ f (E). Sufficiency : Let h be a one-to-one corres-pondence between X and Y such that for each EerX p is a limit point of E i f and only i f q is a limit point of hCEl. Let X = { x.j , Xg, .... } and Y = {h( X l), h(x 2), } then for any n £ N , there is a homeomorphism h n of N * onto N * such that h n(x n) = Mx n). (x n and h ( x n ) a r e p-points)(By Theorem 3-1.3) X and Y are discrete, so there are families {HV>, fK^> of disjoint open-closed subsets of N where x n e H n and h(x n) e Vn eN. Define IT = H^nhjj 1 Qy. Clearly x ne IT and lU^> are disjoint open-closed subsets of N and so are t n n ( u n H • By Theorem 3-1.8, there is a homeomorphism h' of N * onto N * such that h»(U«) = h^IT) and h'(xn) = n n ( x n ) = k^n)* N o w i f u ' i s open-closed set in N containing p, then p i s a limit point of XflU 1 and q i s a limit point of h'(XnU'), hence qeh'CU1). So we have h'(p)=q. .10 Corollary : For any countable infinite set X of P-points * c of N , X has 2 limit points. The set of orbits of * -ft the group of homeomorphisms of N 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 C ( Theorem 0-2. Thus, X has 2° - tf0 = 2 C limit points. There are only c permutations of X, so each u l t r a f i l t e r on X is 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° distinct types of limit points of X; then the set of orbits of the group of homeomorphisms of N onto N has cardinality 2 C. 91 CHAPTER I V HOMEOMORPHISMS OF 3^N INTO ftN 92 4-1.1 Lemma : If f i s a mapping of a non-empty set X into X such that fx = x for no x e X, then there exist d i s j o i n t sets X 1, X2, X^ such that X 1U X 2 U X^ = X and t'ix^l A X ± = 0 1=1,2,3. Proof :-Let us define 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 m a = f n b for some m,n eN (J {o} with f°a = a f o r a l l a e X. cle a r l y R i s an equivalence r e l a t i o n and X i s partitioned by R, f maps each equivalent class of R into i t s e l f ; so i t i s s u f f i c i e n t to prove the theorem for"a set X such that for a l l x,y eX, there exist m,n e N U {o} such that f m x = f n y . Now choose aeX. For any x e X, denote the least m e N U {o] such that f m a = f n x for some n e N U {o} by m(x); denote the least n « N U {oj such that f n x = f m ^ x ^ a by n(x). For any x e X with n(x) > 0 fm(x) a = f n ( x ) x = f n ( x ) - 1 ( f x ) so m(fx) = m(x) and n(fx) = n(x) -1 so m(fx) + n(fx) = m(x) + n(x) -1 . For any x, e X with n(x^) = 0 that i s 93 f^Va = X 1 = f°x 1 fm ( X l ) + 1 a = either (1) m(f X l ) = m(x1) +1 n ( f X l ) = n ( X l ) = 0 Therefore:-m(f X l) + n ( f X l ) = m( X l) + n ( X l ) + 1 or (2) m(f X l) £ m( X l) + 1 that is 3 k(X.j ) such that 0 £ k( X l)< m( X l) + 1 with m(f X l ), = k ( X l ) f k ( X l ) a = f n ( f X l ) ( f X i ) . We claim that such an X^ is unique : Let x,y be distinct points of X with the above properties. Let b = m(x), c = m(y) then „ , „o „b x = f x = f a f X = f b + 1 a = f h a with b+1 >h ^ 0 and y = f°y = f c a fy = f c + 1 a = f k a with c+1 >k = 0 without loss of generality, we may assume b > c. Clearly b > c + 1 = f b a = f b ~ c ( f c a = f b " c - 1 ( f k a ) = f b - ( c + 1 - k ) . x f b ~ c ( f c a ) = fb-°y = f b - c " 1 ( f y ) 94 Since b > c+1 > k b > c+1 - k > 0 Therefore b >b-(c+1-k) ^  0 . This leads to a contradiction to the d e f i n i -t i o n of b. If such an exists, put X 1 = ixj Otherwise = 0. Then m(fx) + n(fx) = m(x) + n(x) - 1 or m(fx) + n(fx) = m(x) + n(x) + 1 Whenever x € X - X^  Now, l e t Xp consist of a l l x e X - X^  with m(x) + n(x) even and X^ consist of a l l x eX - X^  with m(x) + n(x) odd. It i s clear that L U X , U X , = X and 1 2 t> f i x ^ n x . = 0 i = 1 , 2 , 3 . 4 - 1 . 2 Lemma : If D i s a discrete space, and f: D —»D, then the set of fixed points of f, the extension of f to a mapping of j3D to f3D, i s the closure of the set of fixed points of f. Proof :-Let X Q be the set of fixed points of f, then by Lemma 4 - 1 . 1 , 3 d i s j o i n t sets X^X^, X^ such that X. U X 0 U X, = D - X and fCX.] OX, =0 for 1 2 3 o 1 1 ^ i = 1 , 2 , 3 . 95 Now cl e a r l y G l PDXo U G 1 PD X1 U 0 1 {>DX2 U G l p X 3 = P D and 01 p DX Q, 01 (gI)X1 , Gl p DX 2, Gl are d i s j o i n t open-closed sets i n since X Q, X^  , X 2, X^ are di s j o i n t zero-sets i n D. Since f i s continuous and D i s dense i n (3D, the result follows. 4-1.3 Theorem : If f i s a homeomorphism of /3N onto i t s e l f , and i f P i s the set of a l l fixed points of f, then P = G I p N ( P n N ) Proof :-This follows from the fact 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 : If f i s a homeomorphism of /3N into N , then f has no fixed point. Proof :-(1) For every k e N there exists G^c N such that ( i ) G,HG. = 0 whenever j^k, j e N and £ D k^N Gk = N -( i i ) k ^ G k ( i i i ) ffk> U G^ i s a neighborhood of fk i n N U f CN3 . 96 An example of ^ &k: keN} that s a t i s f i e s a l l the three conditions : Let us construct {F^: kcN } by induction as follows : (a) F ^ We have f(1)eN*. Since J1) i s open i n pN, f ( l ) i s open i n f[/JNl therefore there exists c N such that Q OfCjJN] = ff(1 )> where Q i s Aj with A 1 c N. Let F 1 = A1 - {1} . (b) F 2: Similarly we have f(2) e N* and Q 2cK* such that Qg n f E^ NT = (f(2)j where Q 2 i s A£ with A 2 c N. Clearly {f(2)}c pN - C l p N F 1 = CI N ( N - F^ ) Let F 2 = (A 2n N - F 1 ) - {2} . (c) F..: when f F ^ : i < j } i s defined: 3 Q . c l * such that J Q. f)£lpm = |f(j)} where J Q, i s Al with A.c N. (f(d)) cPN - ^  C l ^ F . = °V N - . i ^ F i } Let F j = (AjHN - ^ 1 ^ ) - f d } • Now we have formed a sequence of di s j o i n t sets {F^: i e N ) such that i 4 F j L. 97 L e t N - . ^ F i = M. Set G1 = F G± = F ± i f i - 1 4 M G i = F i U f 1 " 1 * i f i ' 1 € M * Then we have ( G k : k t N ) s a t i s f y i n g c o n d i t i o n s ( i ) and ( i i ) c l e a r l y . F o r ( i ) • {fk}UGk i s a neighborhood o f f k i n I U f I N ] : By the c o n s t r u c t i o n o f G k, we have fk e 01 p ] JG k G 1 6 N G k C G 1 P K A k U <k-1' • T h e r e f o r e , G l p N G k n f I N ] = f f k } . C l p N G k (\ IN V f CN] ] = G k U {fk} Si n c e C l p ^ G k i s open i n j3N, G k U {fk} i s open i n N 0 f IN3". (2) F o r every n 6 G k, l e t hn = f k , t h e n h i s a mapping o f N onto f CN3.. Set g = f ~ ^ h , t h e n gn ^ n Y n e N and g maps N onto N. Thus g, the e x t e n s i o n o f g t o a mapping o f 0N onto j3N, has no f i x e d p o i n t . S i n c e f k £ C l G k i n N U f I N 3 and hn = f k f o r a l l n e G k, h ( f k ) = f k where h . i s the e x t e n s i o n o f h t o a mapping o f PN i n t o N ; hence hx = x i f x £ f i"N3. # Now suppose t h e r e i s an x e N such t h a t f x = x. Then h ( f x ) = f x = x; so x i s a f i x e d p o i n t o f g . T h i s i s a c o n t r a d i c t i o n . 98 4-1.5 Theorem : If f is a homeomorphism of (JN into i t s e l f , and i f P is the set of a l l fixed points of f, then P = c i p N ( p n N ) . Proof :-Letx be a fixed point under f. (1 ) fCNI - N 4 0 : If fCNI - N = 0, then flN : N —• N, and the result follows from Lemma 4-1.2. So we have to consider the case for fCN1 - N 4 0 only. (2) N - fINI is infinite : 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 in pN, so f(n.) is isolated in J J fC^NT. Therefore there is an infinite A<=N such that ClgjjA nf cpm = ff(n J)} , Cl p NAnfCN] = \f(n.)} A Of CN] =0 and therefore N - f CN] sA i s infi n i t e . (3) x e C l ^ (NflfTN]) : Let S1 = NO f CNJ, S 2 = f [NT - N, M 1 = f" 1 cs^:, M2 = f - 1 [S2T . Clearly, Mj A M2 = 0 and M 1 U M2 = N. Thus M^, have d i s j o i n t , complementary closures i n |3N; so f l M ^ l , f m 2 l have d i s j o i n t closures i n fEjJNl. S 2 = f CM23 C N * and 0N i s not homeomorphic to N , Therefore f£CT p N % N* and f.E C l ^ MgJ i s compact, so i s closed i n N . Hence we have N - fCC1^ N M,,] non-empty open i n N ; so there i s an i n f i n i t e A c N such that A'c N* - flClpjj M21. Since A i s countably i n f i n i t e , we can decompose A into countably many d i s j o i n t i n f i n i t e sets A n (ne N). The sets A^ are then d i s j o i n t , open-closed i n N . Now we define a homeomorphism g of N into N such that g(n) = a n 6 A n i f n e ^  , g(n) = f(n) i f n e M2 . Now, (g(n): n £ N } i s a countable i n f i n i t e discrete subset of N . By Theorem 3-1.7, g, the extension of g to |8N into N i s a homeomorphism of |8N into N . By Lemma 4-1 .4, g has no fixed point. But glM 2 = f l M 2 so gloipj M2 = f IClpjj M2. 100 Therefore, x 4 01^ M 2« Thus x = fx 4 01 f tp N J (fENI -N) which is •in-compact, so is closed in N , and so is closed in (3K. We have x4 01^ (fCNI - N) , then x e C l ^ (fXNl n N) , 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 is m eM1 of (3) above ), and h(m) £ m whenever f(m) t fINI fl N. h, the continuous extension of h to pN into £N is a homeomorphism of 3N onto hE|5N3 and the set of fixed points of h is the set of fixed points of f : (i) If x i s a fixed point under f, then x = f(x). x eci^(f CNJHN) by (3), that is f(x) € C l ^ U CNin N), so x € f ~ 1 C l p N ( f CNJ f\ N) = C l ^ N f " 1 (f INI 0 N). By definition, h(m) = f(m) whenever f(m) e fCNlfiN, that is h(m) = f(m) whenever m e f [fINIn NJ. So h o f~ 1 is the identity mapping in f ENIA N, and therefore h . f" 1 is the identity mapping in ClpjjEf EN] ft N] . So x is also a fixed point under h. ( i i ) If y i s a fixed point under E, E(y) = y then y e C l ^ f " 1 Cf CNJ 0 NJ : f" 1 it CN) H NI and f" 1 If IN] - NJ are disjoint, complementary subsets of N; so are their closures in 0N. By theorem 4-1.2, and by definition of h that h(m) £ m whenever f (m) 4 f i m HN, that is whenever m e f~1'Cf CN] fl NI, we have y e C l p N f " 1 If INI 0 NI = f"" 1Glp N(fIN] ON) As in (i) above, E • f~ is the identity mapping in C l ^ I f EN1 fl NJ, therefore f(y) = h(y) = y. So y is also a fixed point under f. Definition of h in (4) : h(m) = f(m) i f m e f" 1[fIN] n N3. Since N - fCNJ is infinite ((2) above). N - f" 1 CflNIONJ i s at most countable, let h(m) m be defined such that h(m)e N - flNJ; for a l l me N - f 11fCNlftNJ. P is the set of a l l fixed points of f. Therefore P is the set of a l l fixed points of h. (By (2) above), and E is the extension of h to a mapping of j3N to |3N. By Lemma 4-1.2, P is the closure of the set of fixed points of h. By definition of h, the set of fixed points of h is the set of a l l fixed points 102 of fIN, that is PON. Therefore P = G l ^ ( P fl N ). 4-1.6 Corollary : If f is a homeomorphism of ^ N into i t s e l f , and i f P^ is the set of a l l k-periodic points of f, then P k = C l p ( P k f l N ) . 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 is 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 are particular cases of Theorem 4-1.5. BIBLIOGRAPHY 103 [1] N. Boboc and GH. Siretchi, Sur l a compactification d^un espace topologique. Bull. Math. Soc. Math. Phys. R. P. Roumaine (N. S.) 5(53) (1961) 155-65(1964). [2] Z. Frolik, Fixed points of maps of /8N. Bull. 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 8(1967), 431-433. [6J J. Kelley, General Topology. Van Nostrand N.Y. (1961). [7] K. D. Magill, Jr., 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 Ul 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 in the Theory of Cech Compactification. Duke Math. J. 23(1965), 409-420. 

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