ADMISSIBLE SUBRINGS OF REAL-VALUED' CONTINUOUS FUNCTIONS by ENG-UNG CHOO B.Sc. Nanyang University, Singapore. 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mathematics We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA August, 1971 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 make 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 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 Supervisor : Professor J.V. Whittaker Abstract The object of t h i s thesis i s to study the r e l a t i o n s between the a l g e b r a i c properties of admissible subrings [ D e f i n i t i o n 0.4] of the r i n g C(X) of a l l real-valued continuous functions on X and the t o p o l o g i c a l properties of the space X. Given an admissible subring G of C(X), there e x i s t s a unique G*-compactification [ D e f i n i t i o n 1.7 and 0.17] Y of X, with the properties that X i s G*-embedded [ D e f i n i t i o n 0.16] i n Y and G*(Y) [ D e f i n i t i o n 0.12] i s admissible. I f the c a r d i n a l i t y |Y — X| of Y - X i s f i n i t e or dG [ D e f i n i t i o n 2.1] i s f i n i t e , then jY — X| = dG. From Y, we can get the unique G-realcompactification [ D e f i n i t i o n 1.8] Z of X where X i s G-embedded i n Z, G(Z) i s admissible and Z i s G(Z)-realcompact [ D e f i n i t i o n 1.1]. I t i s proved that a G-realcompact space X and an Hrrealcompact space Y are homeomorphic i f G and H are isomorphic admissible subrings. Let D(X) be the subring of a l l closed bounded functions i n C(X). Then X i s countably compact i f f D(X) i s admissible and closed under uniform convergence. For any admissible subring G of C(X), i f dG i s f i n i t e , then dG <_ dD(X). Let B(X) be the subring of a l l bounded functions i n C(X) which niap zero sets to closed sets i n R. Then X i s pseudocompact i f f B(X) i s admissible and closed under uniform convergence. TABLE OF CONTENTS Page INTRODUCTION 1 CHAPTER 0 : PRELIMINARIES 3 CHAPTER 1 §1 : G-compactness and G-realcompactness 8 §2 : G*-compactification and G-realcompactification 10 §3 : Isomorphisms and Homeomorphisms . 20 §4 : C h a r a c t e r i z a t i o n of Maximal Ideals of G 25 CHAPTER 2 §1 : dG and sG 29 §2 : Existence of G n with dG n = n, • s G n = n 31 §3 : Equivalent D e f i n i t i o n s of sG and dG. 32 §4 : The G*-compactification and dG 39 CHAPTER 3 §1 : D(X) 41 §2 : D(X) and Countable Compactness 45 §3 : Upper Bound of dG 46 §4 : B(X) 47 §5 : B(X) and Pseudocompactness 50 §6 : S u f f i c i e n t Conditions f o r B(X) = D(X) 51 §7 : Examples 54 BIBLIOGRAPHY 57 ACKNOWLEDGEMENTS I am greatly indebted to Professor J.V. Whittaker f o r h i s valuable suggestions and generous assistance during the preparation of t h i s t h e s i s . I also l i k e to thank Dr. K.Y. Lam f o r h i s constructive c r i t i c i s m of the draft form of th i s work. The f i n a n c i a l support of the U n i v e r s i t y of B r i t i s h Columbia and the National Research Council of Canada i s g r a t e f u l l y acknowledged. Introduction Let C(X) denote the r i n g of real-valued continuous functions on an a r b i t r a r y t o p o l o g i c a l space X. Let C*(X) denote the subring of a l l bounded functions i n C(X). In [1], Gillman and J e r i s o n gave a systematic study of C(X) and C*(X). The i n t e r r e l a t i o n s between C*(X), C(X), the Stone Cech compactification 8X of X and the r e a l c o m p a c t i f i c a t i o n vX of X have been explored to a great extent. They found a nice way to construct BX and vX from C*(X) and C(X). This l e d to the conclusions : (1) BX i s characterized as that compactification of X i n which a l l the bounded real-valued continuous functions on X are extendable over 8X. (2) vX i s characterized as that r e a l c o m p a c t i f i c a t i o n [ D e f i n i t i o n 1.8] of X i n which a l l the real-valued continuous functions on X are extendable over vX. The object of t h i s t h e s i s i s to study the r e l a t i o n s between the alg e b r a i c properties of some s p e c i a l subrings of C(X) and the t o p o l o g i c a l properties of the space X. In p a r t i c u l a r , we define two s p e c i a l subrings D(X) [ D e f i n i t i o n 3.1] and B(X) [ D e f i n i t i o n 3.9] which say a great deal about the structure of the space X. In Chapter 0, we quote some simple lemmas without proofs and some r e s u l t s i n [1] which are used i n the t h e s i s . In Chapter 1, we generalize the theory i n [1]. Given a subring G of C(X), G-compactness [ D e f i n i t i o n 1-1] and G-realcompactness [ D e f i n i t i o n 1.1] have been defined as a nat u r a l g e n e r a l i z a t i o n of compactness and realcompactness r e s p e c t i v e l y . I f G i s admissible [ D e f i n i t i o n 0.4], the G*-compactification of X [ D e f i n i t i o n 1.7] can be obtained i n the same way as BX i s obtained from C*(X). The G*-compactification of X i s charactlzed as that c o m p a c t i f i c a t i o n Y of X where X i s G*-embedded [ D e f i n i t i o n 0.16] i n Y - 2 -and G*(Y) [ D e f i n i t i o n 0.12] i s admissible. From Y, we can obtain the G-realcompactification Z of X [ D e f i n i t i o n 1.8]. Now, Z i s characterized as that r e a l c o m p a c t i f i c a t i o n of X where Z i s G(Z)-realcompact [ D e f i n i t i o n 1.1], X i s G-embedded i n Z and G(Z) i s admissible. Most of the r e s u l t s concerning C*(X), C(X), 3X and vX i n [1] are s t i l l v a l i d when C(X) i s replaced by admissible subring G of C(X). I t i s proved that a G-realcompact space X and a H-realcompact space Y are homeomorphic i f G and H are isomorphic admissible subrings. In Chapter 2, we associate two s p e c i a l numbers sG and dG [ D e f i n i t i o n 2.1] to each subring G of C(X). sG = sup{n : f o r some g i n G, g ^ ( r ) i s not countably compact f o r at l e a s t n r e a l numbers r} and dG = sup{n : f o r some g i n G, g (r) i s not compact f o r at l e a s t n r e a l numbers r}. We give several equivalent d e f i n i t i o n s of sG and dG. If G i s an admissible subring of C(X) and dG i s f i n i t e , then |Y - X| = dG where Y i s the G*-compactification of X. Also, i f |Y - x| i s f i n i t e , then dG = |Y — Xj-In Chapter 3, two s p e c i a l subrings D(X) and B(X) of C*(X) have been studied. Here, D(X) i s the r i n g of a l l closed bounded functions i n C*(X) and B(X) i s the r i n g of a l l functions i n C*(X) which map zero-sets of X to closed sets i n R. We prove that a space X i s countably compact i f f D(X) i s admissible and closed under uniform convergence. Also, a space X i s pseudocompact i f f B(X) i s admissible and closed under uniform convergence. For any admissible subring G of C(X), i f dG i s f i n i t e , then dG <_dD(X). Therefore, dD(X) serves as an upper bound of a l l f i n i t e dG where G i s admissible subring of C(X). Chapter 0 In this Chapter, we set forth some conventions in notation and terminology, and record some preliminary results without proofs. We also quote some well-known results from [1]. Most of the terminology is taken from [1]. A l l topological spaces are assumed to be Hausdorff and completely regular. For any real number r, r also denotes the constant function of value r. The constants of C(X) will then be the constant functions in C(X). For a subset A of a space X, the closure of A in X is denoted by Cl^A. For any subring G of C(X), a l l ideals of G are assumed to be proper ideals of G. 0.1 Definition : A subring G of C(X) is said to determine the topology of the space X i f G seperates the points and the closed subsets of X. 0.2 Definition : Let G be a subring of C(X). A zero-set Z of X is said to be a G-zero-set i f Z = Z(g) for some g in G. 0.3 Definition : A family F of G-zero-sets is said to be a G-filter i f the following hold : (1) * t F . (2) for every Z-^, Z 2 in F, there exists Z-j in F such that z 3c z± n z 2 . (3) i f Z is a G-zero-set which contains some Z' in F, then Z e F. - 4 -A maximal G - f i l t e r is called a G-ul t r a f i l t e r . 0.4 Definition : A subring G of C(X) is said to be admissible i f the following hold : (1) G contains a l l the constants. (2) G determines the topology of X. (3) for every g in G and every real number r, G contains the function (g A r) v(-r). (4) for any non-negative functions f, g in G which have disjoint zero sets, -r—7— e G. f + g 0.5 Lemma : Let F be a G- f i l t e r and L be the collection of a l l functions i n G where Z(f) e F. Then L is an ideal of G. 0.6 Lemma : Let G be a subring of C(X) which is closed under the inversion of functions with empty zero set. Let L be an ideal of G. Then the collection of a l l L-zero-sets i s a G - f i l t e r . 0.7 Example : Let X = R - {0} and G = C*(X). Let L be the ideal of G generated by the identity function on X. Then the collection of a l l L-zero-sets i s not a G-fi l t e r since i t contains the empty set. - 5 -0.8 Lemma : A G - f i l t e r F i s a G - u l t r a f i l t e r i f f F contains a l l the G-zero-sets which meet every set i n F. 0.9 Lemma : Let G be an admissible subring of C(X). Then f o r every g i n G and every closed i n t e r v a l [a;b], g ^([a;b]) i s a G-zero-set. 0.10 Lemma : Let G be an admissible subring of C(X). For every closed subset A of X and a point x i A, there e x i s t s g i n G such that (1) g(x) = 1 and Z(g) 3> A. (2) 0 £ g £ l . 0.11 D e f i n i t i o n : Let G be a subring of C(X) and A be a subset of X. G(A) denotes the c o l l e c t i o n of a l l functions i n C(A) which are r e s t r i c t i o n s of functions i n G. Obviously, G(A) i s a subring of C(A). 0.12 D e f i n i t i o n : Let A be a subset of X and G be a subring of C(A). G(X) denotes the c o l l e c t i o n of a l l functions i n C(X) which are the extentions of functions i n G. Obviously, G(X) i s a subring of C(X). 0.13 Lemma : Let A be a subset of X. For every subring G of C(X), G(A)(X) =? G. 0.14 Example : Let X be the d i s c r e t e space of a l l integers and A be the set of a l l even integers. I f G = C*(X), then G(A)(X) i s the c o l l e c t i o n of a l l functions i n C(X) which are bounded on A. Thus, G(A)(X) 2 G and G(A)(X) * G. - 6 -0.15 Lemma : Let A be a subset of X. I f G i s an admissible subring of C(X), then G(A) i s admissible. The converse i n not true. 0.16 D e f i n i t i o n : Let A be a subset of X and G be a subring of C(A). A i s sai d to be G-embedded i n X i f f G ( X ) ( A ) 3 G. 0.17 D e f i n i t i o n : For every subring G of C(X), G* denotes the i n t e r s e c t i o n G fl C*(X). 0.18 Lemma : Let A be a subset of X and G be a subring of C(X). Then G(A)* 2 G*(A). I f A i s dense i n X, then G(A)* = G*(A). 0.19 Notations ( i ) 6X denotes the Stone-Cech compactification of X and vX denotes the re a l c o m p a c t i f i c a t i o n of X i n which X i s C-embedded. For every f i n C*(X), o f i s the extension of f over BX. ( i i ) For every p i n BX, M p = {f e C(X) : p e C l o v Z ( f ) } P A and M*p = {f e C*(X) : f 6 ( p ) = 0} . 0.20 Theorem : (Gelfand-Kolmogoroff) : The maximal i d e a l s of C(X) are p r e c i s e l y the sets M p where p e BX. 0.21 Theorem : The maximal i d e a l s of C*(X) are p r e c i s e l y the sets M*P where p e BX. - 7 -0.22 Theorem : X i s pseudocompact i f f i t contains no C-embedded copy of N where N i s the d i s c r e t e space of p o s i t i v e i n tegers. 0.23 Theorem : I f A i s a C-embedded subset of X, then i t i s completely seperated from every zero-set d i s j o i n t from i t . 0.24 Notation : For every p i n BX, A p = (Z(f) : p e C l Z ( f ) } . Chapter 1 In [ 1 ] , G i l l m a n and J e r i s o n gave a s y s t e m a t i c s tudy of the i n t e r r e l a t i o n s between C * ( X ) , C ( X ) , BX and v X . In t h i s C h a p t e r , we r e p l a c e C(X) by any a d m i s s i b l e s u b r i n g of C(X) and develope a p a r a l l e l t h e o r y . Most of the techniques used a r e s i m i l a r to those used i n [1]• I t i s a n a t u r a l g e n e r a l i z a t i o n of the theory i n [ 1 ] . S e c t i o n 1 : G-compactness and G - r e a l c o m p a c t n e s s . 1 .1 D e f i n i t i o n : L e t G be a s u b r i n g of C ( X ) . X i s s a i d to be G-compact i f f G has no f r e e i d e a l s . X i s s a i d to be G-rea lcompact i f f G has no f r e e r e a l maximal i d e a l s . 1.2 Theorem : Suppose G i s an a d m i s s i b l e s u b r i n g of C ( X ) . Then X i s G-compact i f f X i s compact. P r o o f : I f X i s not compact, then C(X) has a f r e e maximal i d e a l M . L e t L = M O G. Then L i s an i d e a l of G . F o r every x i n X , t h e r e e x i s t s f i n M such t h a t f ( x ) ^ 0 . S ince G i s a d m i s s i b l e , hence t h e r e e x i s t s g i n G such tha t g(x) = 1 and Z(g) c o n t a i n s Z ( f ) . Now M i s a maximal i d e a l of C ( X ) , Z(g) c o n t a i n s Z ( f ) and f i s i n M . T h e r e f o r e , g i s i n M. Thus g i s i n L and g(x) ^ 0 . T h i s shows t h a t L i s a f r e e i d e a l of G. Hence X i s not G-compact. - 9 -Next, assume that X i s compact. Let L be an i d e a l of G. Suppose there e x i s t s f i n L such that Z(f) = <)>. Since X i s compact, hence there e x i s t s 6 > 0 such that f - 6 i s non-negative. By the a d m i s s i b i l i t y of G, i s i n G. Therefore, <S = —j ' ( f ) 6 + ( f 2 - 6) . I f J i s i n L. I t follows that L = G. This i s a c o n t r a d i c t i o n . Thus, a l l functions i n L have non-empty zero-sets. Hence L i s contained i n some i d e a l M of C(X). M i s f i x e d since X i s compact. Therefore, L i s f i x e d . Consequently, X i s G-compact. 1.3 Theorem : Let G be an admissible subring of C(X). If X i s G-realcompact, then X i s realcompact. Proof : Let M be a r e a l maximal i d e a l of C(X). Then M = M p f o r some p i n vX. Let ([> be the homomorphism from C(X) onto R defined by <J>(f) = f*(p) f o r every f i n C(X). Then M p = kernel <j>. M p H G i s thus a r e a l maximal i d e a l of G. Since X i s G-realcompact, hence Mp H G i s a f i x e d i d e a l . That i s , there e x i s t s x 0 i n X such that g(x Q) = 0 f o r every g i n Mp H G. Assume that x Q ^ p. Let Z be a zero set i n A P such that x D i s not i n Z. Since G i s admissible, hence there e x i s t s g i n G such that g(x c) = 1 and Z(g) contains Z ( f ) . Since Z(g) contains Z ( f ) , f i s i n M p and M p i s a maximal i d e a l of C(X), hence g i s i n Mp. Thus g i s i n M p n G and g(x Q) ^ 0. This i s a c o n t r a d i c t i o n . Hence p = x Q e X. Therefore, M P i s f i x e d . - 10 -Consequently, X i s realcompact. Example : Let G = {f e C(R) : there e x i s t s n such that f i s constant on R - (-n;n)}. Then R i s not G-realcompact. But R i s a realcompact space. Proof : Let <f> be the mapping from G to R defined as follows : <j)(g) = l i m g(n) , f o r every g e G. n-x» Then (J> i s an epimorphism. The kernel of <j> i s a free r e a l maximal i d e a l of G. Hence R i s not G-realcompact. Section 2 : G*-compactification and G-realcompactification. 1.4 Lemma : Let X be a subset of a space Y and G be an admissible subring of C*(X) such that G(Y) i s admissible. Suppose every point y of Y i s the l i m i t of a unique G - u l t r a f i l t e r A^. Then f o r every y i n Y, A? i s the c o l l e c t i o n of a l l G-zero-sets whose closures i n Y contain y-Proof : Let Z be a G-zero-set i n A^. Since A^ converges to y, hence Z n V £ <j> for every neighbor-hood V of y i n Y. Therefore, y e ci Yz. Conversely, l e t Z be a G-zero-set such that y e Cl^Z. Let E be the c o l l e c t i o n of a l l sets of the form F (\ X where F i s a G(Y)-zero-set neighborhood of y i n Y. Then E U {Z} has the f i n i t e i n t e r s e c t i o n - 11 -property. Since every G - f i l t e r containing E converges to y. Hence, by hypothesis, A y contains E U {Z}. Thus Z e A y. This completes the proof. 1.5 Lemma : Let X be a subset of a space Y and G be an admissible subring of C*(X) such that G(Y) i s also admissible. If every point y of Y i s the l i m i t of a unique G - u l t r a f i l t e r A y, then X i s G-embedded i n Y. Proof : Let g be a function i n G. Then there e x i s t s a p o s i t i v e integer n such that g(x) CZ [-n;n]. For every y i n Y, l e t B y be the c o l l e c t i o n of a l l sets of the form A H [-n;n] where A i s a closed subset of R and g ''"(A) i s i n A y. Then B y has the f i n i t e i n t e r s e c t i o n property. Since [-n;n] i s compact, hence the i n t e r s e c t i o n of a l l sets i n B y i s not empty. Let g*(y) be a r e a l number i n th i s non-empty i n t e r s e c t i o n . For every 6 > 0, l e t W = [-n;g*(y) - 6] U [g*(y) + 6;n]. Then g - 1 ( [ g * ( y ) - 6;g*(y) + 6]) U g - 1(W) = X. A y i s a G - u l t r a f i l t e r and X i s i n A y. Thus g ([g*(y) - 6;g*(y) + 6]) i s i n A? or g (W) i s i n A y. Since g*(y) i W, hence by lemma 1.4, g ^(W) i A y. Therefore, g _ 1 ( [ g * ( y ) " <S;g*(y) + 6]) i s i n A y. I t follows that [g*(y) - 6;g*(y) + 6] i s i n B y f o r every 6 > 0. Thus g*(y) i s uniquely determined by A y. Therefore, g* i s a well-defined mapping from Y in t o R. - 12 -Since g~ 1([g*(y) - <S;g*(y) + 6]) U g - 1(w) = X, hence Y -Cl y[g~ 1(W)] i s a subset of C l Y [ g ~ 1 ( [ g * ( y ) - 6;g*(y) + 6 ] ) ] . For every z i n C l Y [ g " 1 ( [ g * ( y ) - 6;g*(y) + 6 ] ) ] , g _ 1 ( t g * ( y ) - 6;g*(y) + 6]) e A Z. Thus g*(z) £ [g*(y) - 6;g*(y) + 6]. Hence g*(Y - CLytg'V) ]) C [g*(y)-6;g*(y)+S]. Furthermore, Y - CLy[g ^"(W)] i s a neighborhood of y i n Y. Consequently g* i s continuous at y. Therefore, g* i s continuous. For every x i n X, A x i s the c o l l e c t i o n of a l l G-zero-sets which contain x. Hence B x i s the c o l l e c t i o n of a l l sets of the form A n [ -n;n] where A i s closed i n R and g(x) e A. Therefore, g*(x) = = g(x). Consequently, g* i s the extension of g over Y. 1.6 Theorem : Let X be a dense subset of a space Y and G be an admissible subring of C*(X) such that G(Y) i s admissible. Then the following are equivalent : (1) X i s G-embedded i n Y. (2) d i s j o i n t G-zero-sets have d i s j o i n t closures i n Y. (3) f o r every f, g e G, C l y [ Z ( f ) (\ Z(g)] = ( C l y Z ( f ) ) f \ ( C l y Z ( g ) ) (A) every point of Y i s the l i m i t of a unique G - u l t r a f i l t e r . Proof : (1) — > (2). Let Z ( f ) , Z(g) be two d i s j o i n t G-zero-sets where f, g e G. f 2 Then h = e G and h ( Z ( f ) ) = {0}, h(Z(g)) = {1}. Since X i s f 2 + g 2 - 13 -G-embedded i n Y, hence h i s extendable over Y. I t follows that Z(f) and Z(g) have d i s j o i n t closures i n Y. (2) — > (3). Let y be any point i n ( C l y Z ( f ) ) fl ( C l y Z ( g ) ) since X i s dense i n Y, hence f o r every neighborhood V of y i n Y, y e ( c i y [ v n z ( f ) ] ) n ( c i y [ v n z ( g ) ] ) . By the a d m i s s i b i l i t y of G(Y), there e x i s t s h i n G(Y) such that Z(h) V and Z(h) i s a neighborhood of y i n Y. Then y e ( c i y [ z(h) n z ( f ) ] ) n ( c i y [ z(h) n z ( g ) ] ) . By (2), we have Z(h) H Z(f) H Z(g) ± <j>. Thus V Pi Z(f) (\ Z(g) ^ <j> . Since V i s an a r b i t r a r y neighborhood of y i n Y, hence y e C l y [ Z ( f ) n Z ( g ) ] . Consequently, ( C l y Z ( f ) ) H (CLyZCg)) = C l y [ Z ( f ) H Z(g)]. (3) — > (4). For an a r b i t r a r y point y i n Y, l e t A y be the c o l l e c t i o n of a l l G-zero-sets Z such that y e CLyZ. By (3), A y i s a G - f i l t e r . Let W be a G-zero-set which i s not i n A y. Then y i CLyW. Since G(Y) i s admissible, hence there e x i s t s h i n G(Y) such that h(y) = 1 and Z ( h ) : 3 CLyW. Now, h _ 1([-|;-|]) H X i s a G-zero-set i n A y and h_1([4;|-]) H X H W = <(>. Therefore, A y i s a G - u l t r a f i l t e r by lemma 0.8. - 14 -Let V be any neighborhood of y i n Y. Since G(Y) i s admissible, there e x i s t s g i n G(Y) such that Z(g) CI V and Z(g) i s a neighborhood of y i n Y. Then V O Z(g) n x and Z(g) H X i s a G-zero-set i n A y. Thus A? converges to y. Every G - f i l t e r which, converges to y i s contained i n A y. Hence A y i s the unique G - u l t r a f i l t e r which converges to y. (4) — > (1). This follows from lemma 1.5. 1.7 D e f i n i t i o n : Let G be a subring of C*(X). A t o p o l o g i c a l space Y i s sa i d to be a G-compactification of X i f f the following hold : (1) X i s a dense subset of Y. (2) X i s G-embedded i n Y. (3) G(Y) i s admissible. (4) Y i s compact. 1.8 D e f i n i t i o n : Let G be a subring of C(X). A t o p o l o g i c a l space Y i s said to be a G-realcompact-i f i c a t i o n of X i f f the following hold : (1) X i s a dense subset of Y. (2) X i s G-embedded i n Y. (3) G(Y) i s admissible. - 15 -(4) Y i s G(Y)-realcompact. Example : Let G = {f £ C(R) : there e x i s t s an integer n such that f i s constant on R - [-n;n]}. Then G i s an admissible subring of C*(R). The one-point compactification R* of R i s a G-compactification of R. 1.9 Theorem : Let G be a subring of C(X) where G* i s admissible. Then X has a G*-compactification. C* Proof : Furnish P = R with the c a r t e s i a n product topology. Let a be the map from X i n t o P defined as follows : f o r every x i n X, o(x)(g) = g(x) f o r every g i n G*. For every g i n G*, l e t IT denote the p r o j e c t i o n IT (<j>) = <t>(g) f o r every <(> i n P. Since i r B - a = g i s continuous f o r every g i n G*, hence a i s a continuous map. If a(x) = a(x'), then g(x) = g(x') f o r every g i n G*. Since G* i s admissible, i t follows that x = x'. Thus a i s a one-one map. Hence a 1 i s a well-defined map from o(x) onto x. For every g i n G*, g-a 1 coincides with H g on a(x). Thus g-a ^ i s continuous f o r every g i n G*. Since G* determines the topology of X, hence a 1 i s continuous. - 16 -Consequently, a i s a homeomorphism from X onto (x). For every g i n G*, Cl_g(x) i s compact. By Tychonoff Theorem, II C l R g ( x ) i s compact. Since a(x) C n Clgg(x), hence C l R a ( x ) i s geG* geG* compact. Let H = {g°a 1 : g e G*}. Since G* i s admissible, hence H i s admissible. For every g i n G*, goo X coincides with 7Tg on cr(x). Thus o(x) i s H-embedded i n Cl^a(x) and R ( C l R a ( x ) ) determines the topology of C l ^ a ( x ) . Furthermore, a(x) i s dense i n C l R a ( x ) . Therefore, H ( C l R a ( x ) ) i s admissible. Consequently, C l R o ( x ) i s a H-compactification of a ( x ) . This completes the proof. 1 .10 Theorem : Let A be a subset of X and G be a subring of C(A) such that G(X) i s admissible. Then A i s G*-embedded i n X i f f CLyA i s a G*-compactification of A where Y i s a G(X)*-compactification of X. Proof : I f A i s G*-embedded i n X, then A i s G*-embedded i n Y. Since G(X)*(Y) i s admissible, hence G*(ClyA) i s admissible. Therefore, ClyA i s a G*-compactification of A. Conversely, suppose Cl^A i s a G*-compactification of A. Then A i s G*-embedded i n C l y A . Since Cl^A i s a compact subset of Y, hence C l y A i s C-embedded i n Y. Thus A i s G*-embedded i n Y. Therefore, A i s G*-embedded i n X. - 17 -1.11 C o r o l l a r y (1) If G = C(X), then 6X i s a G*-compactification of X. (2) Let A be a subset of X. Then A i s C*-embedded i n X i f f C l g x A = BA. 1.12 Theorem : Let G be an admissible subring of C(X) and Y be a G*-compacification of X. Let R* be the one-point compactification of R. Then f o r every g i n G, there e x i s t s a continuous function "g from Y in t o R* which extends g. Proof : For every n, l e t g n = (g An)V(-n) and g^" be the extension of g n over Y. Let y be a point i n Y. We have the following two cases : (1) I^Xy) | < tn f o r some m. By cont i n u i t y of g^, there e x i s t s a neighborhood V of y i n Y such that | gj^Cz) | < m f o r every z i n V. Therefore, g n ^ z ^ = & m( z) ^ o r e v e r v z ^ n V H X and n >_ m. Since y e C L ^ V H X), hence |g^(y) | = = &^(y) whenever n >_ m. In t h i s case, we put g~(y) = "g m(y) • (2) |g^(y)| 21 m f ° r e v e r Y m - Since X i s dense i n Y and |g m| <_ m, hence |g^(y)| = m f o r every m. In th i s case, we put "g(y) = co . It i s obvious that g i s a mapping from Y to R* which extends g. I t remains to show that "g i s continuous. - 18 -Let y be an a r b i t r a r y point of Y. We have to consider two cases : (1) Suppose |g^(y)| < m f o r some m, l e t W be an open neighborhood of y i n Y such that |g^Cz)| < m f o r every z i n W. Then g^(z) = g^( z) whenever n >_ m f o r every z i n W. Therefore, "g(z) = g^(z) for every z i n W. For every e > 0, there e x i s t s a neighborhood V of y i n Y such that g^(V) ^ Cg(y) - E ; g X v ) + £ ) • I t follows that "g(V H W) 9: (g(y) - e;g(y) + e) and V H W i s a neighborhood of y i n Y. Hence "g i s continuous at y. (2) Suppose | g^(y) | = n f o r every n. Then g~(y) = oo . For every compact subset K of R, there e x i s t s n such that K ^ [-n;n]. Since |g" n^(y) | = n + 1, hence there e x i s t s a neighborhood U of y i n Y such that |s"n+l^^l > n f o r every z i n U. Therefore, g~(U) C R* - K. Hence "g" i s continuous at y. Consequently, g~ i s a countinuous function from Y to R*. 1.13 Theorem : Let G be an admissible subring of C(X) and Y be a G*-compactification of X. Let S = {y e Y : g"(y) f oo f o r every g i n G}. Then S i s a G - r e a l c o m p a c t i f - i c a t i o n o f X. Proof : I t i s c l e a r that X i s a dense G-embedded subset of S and G(S) i s admissible. Let (ft be a homomorphism from G(S) onto R and ^ be the homomorphism of G*(Y) onto R defined by : for every g i n G*(Y), ij;(g) = <Kg ' ) where g' i s the r e s t r i c t i o n of g on S. - 19 -Since Y i s compact and G A(Y) i s admissible, hence kernel ty = = kernel ty (\ G*(Y) i s f i x e d . Thus there e x i s t s y c i n Y such that ty(g) = g(y 0) for every g i n G*(Y). Assume that y Q i S. Then there e x i s t s g i n G such that l"(y D) = oo . Let h = g 2 + 1. Then h e G, h >_ 1 and h ( y D ) = oo . Since G i s admissible, hence -\ = -pr \r—-—- e G. Also ^ i s bounded since h (h - 1) + 1 h h > 1. Let ( i ) ' e G(S) and (j^)" e G*(Y) be the extensions of ^ . Then ( i ) " ( y ) = 0 since h ( y 0 ) = oo . Thus <f>((£)') = = (£)"(y 0> = 0. But then 1 = ty (1) = ipCC^'-h') = <f> ((£) ')-ty (h*) = 0 where h* e G(S) i s the extension of h. This i s a c o n t r a d i c t i o n . Hence y e S. I t remains to show that kernel <j> = {g e G(S) : g ( y Q ) = 0}. Let g^ e kernel ty and g = g 2 . Then <fi (g) = 0. Since g i s non-negative, hence g(y Q) ^_ 0. Assume that g ( y Q) = 6 > 0 f o r some 6 . Let m > 6 and gm = (g A m)V(-m). Then g m e G(S)*. Let g* e G*(Y) be the extension of Sm- Then *(g^) = g*(y c) = g m ( y 0 ) = 5 • T h u s <Kgm> = <Kg£) = «• Therefore, g m - 6 e kernel ty. Let h = ( g m - < 5 ) 2 + g2. Then h e kernel ty . Moreover, fi2 1 h >_•--- > 0. Therefore, — e G(S). I t follows that 1 e kernel ty. This i s a c o n t r a d i c t i o n . Hence, g(y Q) = g|(y 0) = 0. Thus kernel ty C {geG(S):g(y o)=0} Since kernel ty i s a maximal i d e a l of G(S), hence kernel <j> = {geG(S) :g(y o)=0} which i s f i x e d . ' This completes the proof. 1.14 C o r o l l a r y : I f G = C(X), then Y = gX and S = vX. - 20 -Section 3 : Isomorphisms and Homeomorphisms. Let G, H be admissible subrings of C(X), C(Y) r e s p e c t i v e l y . Then G* and H* are isomorphic i f f the G*-compactification of X and the H*-compactification of Y are homeomorphic. Also G and H are isomorphic i f f the G-realcompactification of X and the H-realcompactification of Y are homeomorphic. I t follows that the G*-compactification and the G-realcompactification of X are uniquely determined by G and X. 1.15 Lemma : Let G be a subring of C(X) which contains a l l the constants and H be a subring of C(Y) which contians a l l the constants. I f <j) i s a homomorphism from G onto H, then <|) (r) = r f o r every r i n R. Proof : Since <J> i s onto, hence there e x i s t s f i n G such that <f>(f) = 1. But cj>(f) = <j>(f-D = 4>(f)-<KD = 'I'd)- Therefore, <J> (1) = 1. I t follows that <f>(r) = r f o r every r a t i o n a l number r . Let r be any r e a l number. For every e > 0, there e x i s t r a t i o n a l numbers a, B such that r - e < o t < r < B < r + e. Since r - a >_ 0 and 3 - r >_ 0, hence <|>(r - a) = <j> ( ( / r - a) ^ ) >_ 0 and 4>(B - r) = <j>((/6 " r ) 2 ) _> 0. Thus (j> (r) >_ <|>(a) = a and 6 = <f>(6) >_ <J> (r) . Therefore, r - e <_<J>(r) <_ r + E . Since e > 0 i s a r b i t r a r y , hence <f)(r)=r. 1.16 Theorem : Let G, H be admissible subrings of C(X), C(Y) r e s p e c t i v e l y . Suppose X i s G-realcompact and Y i s H-realcompact. If <f> i s an isomorphism - 2 1 -from G onto H, then there e x i s t s a unique homomeomorphism x from Y onto X such that <f> (g) = g' T f o r every g i n G and <(> '''(h) = h-x ^ f o r every h i n H. Proof : Let y Q be a point i n Y. Let y Q be a mapping from G i n t o R defined by : y 0 ( g ) = <Kg)(yo) f o r every g i n G. Then y Q i s a homomorphism. Since y 0 ( l ) = 1 , hence kernel y Q i s a r e a l maximal i d e a l of G. By hypothesis, X i s G-realcompact. Therefore, there e x i s t s x D i n X such that kernel y 0 = {g e G : g(x Q) = 0 } . The point x Q i s uniquely determined by y Q . Let x be a mapping from Y i n t o X be defined as follows : x(y) = x where kernel y = {g e G : g(x) = 0 } , f o r every y i n Y. Let g e G. For every y i n Y, y(g - y(g)) = (J>(g - y(g))(y) = = <|>(g)(y) - y(g)(y) = <f>(g)(y) - <f>(g)(y) = 0 . Hence g - y(g) e kernel y f o r every y i n Y. Thus g(x(y)) = y(g)(x(y)) = y(g) f o r every y i n Y. That i s , (g«x)(y) = <f>(g)(y) f ° r every y i n Y. Therefore, g.x=(j>(g). Consequently, <f> (g) = g ' T f o r every g i n G. S i m i l a r l y , <j> ^ i s an isomorphism from H onto G and Y i s H-realcompact, therefore there e x i s t s a mapping a from X i n Y such that <f> "'"(h) = h-a f o r every h i n H. - 2 2 -Since g . T = <j>(g) i s continuous for every g i n G and G determines the topology of x, hence T i s continuous. Since h»a = ty ^(h) i s continuous f o r every h i n H and H determines the topology of Y, hence a i s continuous. For every x i n X and g i n G, we have g[(x«a)(x)] = = [g-x](a(x)) = <f>(g)(a(x)) = [<Kg)«a](x) = [<|>"1(<Kg)) ] (x) = g(x). Therefore, T-a i s the i d e n t i t y map on X. S i m i l a r l y , a-x i s the i d e n t i t y map on Y. Hence T i s a homeomorphism and a = x l . Let T' be a map from Y i n t o X such that <f>(g) = g*x' f o r every g i n G. For every y i n Y, since g(x(y)) = <J> (g) (y) = g(x'(y)) f o r every g i n G and G seperates the points of X, hence x(y) = x*(y), Therefore, T i s uniquely determined by <j>. This completes the proof. 1.17 C o r o l l a r y : Let G, H be admissible subrings of C(X), C(Y) r e s p e c t i v e l y . Suppose X and Y are compact. I f <j> i s an isomorphism from G onto H, then there e x i s t s a unique homeomorphism T from Y onto -1 -1 X such that ty(g) = g . T f o r every g i n G and ty (h) = h-x f o r every h i n H. Proof : X i s G-realcompact and Y i s H-realcompact. The a s s e r t i o n then follows from Theorem 1.16. - 23 -1.18 C o r o l l a r y : Let G be an admissible subring of C*(X). Then the G-compactification of X e x i s t s uniquely up to homeomorphism. Proof : The existence of the G-compactification of X has been proved i n Theorem 1.9. Suppose Y-p Y 2 are G-compactifications of X. For every f i n G(Y^), there e x i s t s a function <j>(f) i n G ( Y 2 ) such that f and <J>(f) coincide on X. I t i s obvious that (> i s an isomorphism from GCY-^ ) onto G ( Y 2 ) . Hence by C o r o l l a r y 1.17, Y^ and Y£ are homeomorphic. 1.19 C o r o l l a r y : Let G be an admissible subring of C(X). Then the G-realcompactification of X e x i s t s uniquely up to homeomorphism. Proof : The existence of the G-realcompactification of X has been proved i n Theorem 1.13. Suppose Yj_, Y 2 are G-realcompactif i c a t i o n s of X. For every f i n G(Y^), there e x i s t s a function <j>(f) i n G ( Y 2 ) such that f and <j>(f) coincide on X. I t i s obvious that <j> i s an isomorphism from G(Y]_) onto G ( Y 2 ) . Y-^ i s G(Y^)-realcompact and Y 2 i s G ( Y 2 ) - r e a l c o m p a c t . Hence by Theorem 1.16, Y-^ and Y 2 are homeomorphic. Examples : (1) Let X = (0;1) and G = {f e C(X) : there e x i s t s a compact subset K of X such that f i s constant on X - K} . Then G i s an admissible subring of C*(X). - 24 -The one-point compactification of X i s the G-compactification of X. (2) Let X = (0;1) and G = {f e C(X) : there e x i s t s a compact subset K of X such that f(X - K) i s f i n i t e } . Then G i s an admissible subring of C*(X). [0;1] i s the G-compactification of X. 1.20 Theorem : Let G, H be admissible subrings of C(X) such that G C H. Let Y be the H*-compactification of X. Then Y i s the G*-compact i f i c a t i o n of X i f f G*. separates., d i s j o i n t H-zero-sets. Proof : Suppose Y i s the G*-compactification of X. Let A, B be two d i s j o i n t H-zero-sets. Since Y i s the H*-compactification of X, X i s H*-embedded i n Y. By Theorem 1.6, CLyA H C l y B = cf>. G*(Y) i s admissible and ClyA, ClyB are d i s j o i n t compact s e t s . Hence there e x i s t s g* i n G*(Y) such that g*(Cl YA) = {0} and g*(Cl yB) = {1}. Let g be the r e s t r i c t i o n of g* on X. Then g e G * and g . s e p a r a t e s A and B. Suppose G* seperates d i s j o i n t H-zero-sets. Since G c: H and X i s H*-embedded i n Y, hence X i s G*-embedded i n Y. To prove that Y i s the G*-compactification of X, i t remains to show that G*(Y) determines the topology of Y. Let A be a closed subset of Y and y i A. Since H*(Y) determines the topology of Y, hence there e x i s t s h* i n H*(Y) such that h*(y) = 1 and A ^ Z(h*). Let h be the r e s t r i c t i o n of h* 1 1 1 — 1 2 A on X. Then h e H*. h~ ( [ - - j ; -j]) and h ([-^J-j]) are d i s j o i n t H-zero-sets. Thus there e x i s t s g i n G* such that g(h L ( [ - 4;-^ ])) = (Ol and - 25 -g(h ([-J;-J])) = {!}• Let g* be the extension of g over Y. Then g* e G*(Y). Since X i s dense i n Y, hence y e C l y [ h ~ ([-j;-^]')] and A C C l Y [ h - 1 ( [ - Therefore, g*(y) = 1 and Z(g*) 5 A. Thus Y i s the G*-compactification of X. Example : Let X be the set of a l l integers with the d i s c r e t e topology. Let G = {f e C(X) : f(x) i s f i n i t e } . Then G i s an admissible subring of C*(X) and G seperates d i s j o i n t zero sets of X. Hence 3X i s the G-compactification of X. Section 4 : C h a r a c t e r i z a t i o n of Maximal Ideals of G. Let G be an admissible subring of C(X). I f G i s closed under i n v e r s i o n of functions with empty zero set, then there e x i s t s a one-one correspondence between the maximal i d e a l s of G and the points i n the G*-compactification of X. 1.21 Theorem : Let Y be the G*-compactification of X where G i s an admissible subring of C(X). I f G i s closed under the i n v e r s i o n of functions with empty zero set, then the maximal i d e a l s of G are p r e c i s e l y the sets M y = {g e G : y e CLyZCg)} , y e Y and they are d i s t i n c t f o r d i s t i n c t y. Proof : Let M be a maximal i d e a l of G. Since G i s closed under the i n v e r s i o n of function with empty zero set, hence every function i n M has non-empty zero set. Let F = {CLyZ(g) : g E M} . Then F has f i n i t e i n t e r s e c t i o n property. Since Y i s compact, hence there e x i s t s y such that y E CLyZCg) f o r every g E M. Thus M My. I t remains to show that M y i s an i d e a l of G. Let f, g be any functions i n M y. Then y e C l y Z ( f ) and y E C l Y Z ( g ) . By Theorem 1.6, CLyZCf) n C ] T Z ( g ) = C l Y [ Z ( f ) H Z(g) ]. Therefore, y e C l ^ Z C f ) ft Z(g)]. Hence y e CL^ZCf - g). Thus f-g e My. For every h i n M y and f i n G, y E C l Y Z ( h ) . Therefore, y e CLyZChf). Thus hf e My. This shows that M y i s an i d e a l of G. M M y and M i s a maximal i d e a l of G. Therefore, M = My. I t i s obvious that M y f M Z i f y ^ z. This completes the proof. 1.22 C o r o l l a r y : p r e c i s e l y the sets Proof : When G Remark : Let X be the set of a l l p o s i t i v e integers with the d i s c r e t e topology. Let G = {f e C(X) : f i s bound on even integers}. Then G i s an admissible subring of C(X). I t i s c l e a r that G* = C*(X). Hence gX i s the G*-compactification of X. Let g be the function i n G defined as follows : I f G = C(X), then the M P =' {g e C(X) : p e = C(X), G* = C*(X) and maximal i d e a l s of C(X) are c i 3 X z ( g ) } , p E gX. Y = gX. - 27 -g(n) = 1 — , i f n i s even n n , i f n i s odd Then Z(g) = <j> and - i G . Let M be a maximal i d e a l of G such that g e M. Since Z(g) = <f>, hence M ^ M P f o r every p e BX. Therefore, i t i s e s s e n t i a l that G i s closed under the i n v e r s i o n of f u n c t i o n with empty zero set i n Theorem 1.21. 1.23 Theorem : Let G, H be admissible subrings of C(X), C(Y) r e s p e c t i v e l y . Let X' be the G*-compactification of X and Y' be the H*-compactification of Y. Suppose <j> i s a continuous function from X i n t o Y such that h-<}> e G* for every h i n H*. Then there e x i s t s a continuous function <j> from X' i n t o Y' which extends the function <j>. Proof : For every p i n X', l e t A P = ( Z ( f ) : f e G* and p e C l v , Z ( f ) } and B P = {Z(h) : h e H* and Z(h-<f.) e A P}. Then {C1 Y,Z : Z e B p} has the f i n i t e i n t e r s e c t i o n property. Since Y' i s compact, hence {Cly,Z : Z e B P} has non-empty i n t e r s e c t i o n . Let q be a point i n the i n t e r s e c t i o n . Denote q by ^"(p). Let V be an open neighborhood of q i n Y'. Since H*(Y) determines the topology of Y', hence there e x i s t s h* i n H*(Y') such that h*(q) = 1 and Y* - V £ Z(h*). Then h* - 1([-oo ;-|]) H Y and - 28 -Y 0 h*~ ([y;co)) are H*-zero-sets. Let f, g e H* such that Z(f) = = h * _ 1 ( [ | ; c o ) ) H Y and Z(g) = h* - 1([-ao ;|]) n Y. Then Z(f) U Z(g)=Y and Y' - CLy.ZCg) C C l Y , Z ( f ) C V. Now, q i C l ^ Z C g ) . Thus Z(g) i B P Therefore, p i C l ,Z(g•<$>). Hence X' - ClvlZ(g-(|>) i s an open neighborho A A of p. Since Z(f) U Z(g) = Y, hence ZCf•*) U Z(g-<f>) = X. Thus Cl x,Z(f-<f.) U Cl^.ZCg-*) = X'. For every x i n X' - Clx,Z(g-<|>) , x e C L ^ Z C f .<J)). Then Z(f) e B x. Since f(x) i s a point i n C l v , Z ( f ) and CLy.ZCf) C V, hence J(x) e V f o r every x s X'-Cl ,Z(g . ( ( i ) . Consequently, we have shown that f o r every open neighborhood V of ^(p) i n Y', there e x i s t s a neighborhood X'-Cl ,Z(g'<}>) of p i n X' such that "$"(X'-Cl x,Z(g •<}>)) C V. T h i s shows that J i s well-defined and i s continuous. I f p E X, then A P = {Z(f) : f e G* and p e Z ( f ) } . Thus B P = {Z(h) : h E H* and <))(p) E Z(h)}. Therefore, ^(p) = <f>(p) • Hence extends the f u n c t i o n <j>. This completes the proof. Chapter 2 Section 1 : D e f i n i t i o n s of sG and dG. 2 . 1 D e f i n i t i o n : For every subring G of C(X), we define sG(X) = sup{n : f o r some g i n G, g ^ ( r ) i s not countably compact f o r at l e a s t n r e a l numbers r} . dG(X) = sup{n : f o r some g i n G, g ^"(r) i s not compact for at l e a s t n r e a l numbers r} . I t i s obvious that sG(X) <_ dG(X). I f X i s a metric space, then sG(X) = dG(X). A space X i s compact i f f dG(X) = 0 for every subring G of C(X). A space X i s countably compact i f f sG(X) = 0 f o r every subring G of C(X). Without ambiguity, we w i l l denote sG(X) by sG and dG(X) by dG. 2 .2 Lemma : Suppose g i s a function i n C(X) where g ^"(r) i s not countably compact ( r e s p e c t i v e l y , compact) i f f r = r j , r 2 , • • • , r m . Let G = {p»g : p i s a polynomial i n C(R)}. Then G i s a subring closed under composition with polynomials i n C(R) and sG = m. ( r e s p e c t i v e l y , dG = m). Proof : Since g i s i n G, hence sG >_ m. Let p«g be any function i n G. I f p has constant value b, then (p*g) ~*~(r) i s not countably compact only i f r = b. Suppose p i s not a constant function. For any r e a l number r, - 30 -l e t O j , a 2 , •••, a n be a l l the r e a l roots of the equation p(x) = r . Then -1 n -1 -1 (p.g) (r) = U g ( a ^ ) . Thus (p-g) (r) i s not countably compact only i = l i t g "^(a^) i s not countably compact for some i . By hypothesis, = r^ fo r some j . Therefore, r = p ( r j ) . Hence (p-g) "''(r) i s not countably compact f o r at most m r e a l numbers r . This shows that sG <_ m. Consequently, sG = m. I t i s easy to see that G i s a subring closed under composition with polynomials i n C(R). The proof of dG = m i s s i m i l a r . 2.3 Theorem : Let G be a subring of C(X). Suppose G contains a l l the constants. Then (a) sG = sup{n : n = 0 or g _ 1 ( l ) , g ^"(2), g *(n) are not countably compact f o r some g i n G} . (b) dG = sup{n : n = 0 or g "'"(l), g 1 ( 2 ) , g 1 ( n ) are not compact f o r some g i n G} . Proof : I t i s obvious that G i s closed under composition with polynomials i n C(R). (a) I f X i s countably compact, the e q u a l i t y i s obvious. Suppose there e x i s t s g i n G such that g (r) i s not countably compact f o r - 31 -r = r 1 , r 2 , • • •, r n . Let p be the polynomial i n C(R) defined by : POO - I f r . - r ) • • • ( r . - r H r . - r ) • • • (r - r ) i = l k r i z l } ^ i r i - l M r i r i + l ; ^ r i r n ; Then p ( r i ) = i for each i = 1, 2, n. Now, p-g e G and -1 -1 —1 (p«g) ( i ) O G (r±) for each i = 1, 2, n. Hence (p-g) (1), (p«g) ^ ( 2 ) , (p«g) ^(n) are not countably compact. Therefore -1 -1 -1 sG <_ sup{ n : n = 0 or g (1), g (2), • • •, g (n) are not countably compact f o r some g i n G }. The reverse i n e q u a l i t y i s obvious. Therefore the e q u a l i t y holds. The proof of (b) i s s i m i l a r . Section 2 : For any subring G of C(X), sG and dG are e i t h e r non-negative integers or oo . Given a non-negative integer k, does there e x i s t a subring G of C(X) such that sG = k or dG = k ? A p a r t i a l answer i s given i n t h i s s e c t i o n . 2.4 Theorem : Let G be a subring of C(X). Suppose G contains a l l the constants and sG = m i s f i n i t e ( r e s p e c t i v e l y , dG = m i s f i n i t e ) . Then there e x i s t subrings G-^ , G 2 , " * " » ^ m 1 °^ ^ such that ( i ) each G^ i s closed under composition with polynomials i n C(R). ( i i ) sG^ = i for each i = 1, 2, m - 1. ( r e s p e c t i v e l y , dG-L=i f o r each i = 1, 2, •••, m - 1). - 32 -Proof : I f m <_ 1, there i s nothing to prove. Suppose m >_ 2. Since sG = m i s f i n i t e , hence there e x i s t s a function g i n G such that g "^(r) i s not countably compact i f f r = 1, 2, m. For each k = 1, 2, m - 1, l e t p^ be a polynomial i n C(R) such that P^Ci) = i f o r each i = 1, 2, k and Pj^(i) = k f o r each i = k + 1, k + 2, •••, m. Let g k = p^-g. Then g k e G and gj^Cr) i s not countably compact i f f r = 1, 2, k. Let G^ = {p'gfe. : p i s a polynomial i n C(R)}. By lemma 2.2, each G^ s a t i s f i e s the conditions ( i ) and ( i i ) . The proof of the a s s e r t i o n about dG i s s i m i l a r . Section 3 : In t h i s s e c t i o n , s e v e r a l equivalent d e f i n i t i o n s of sG and dG are given. 2.5 Theorem : Suppose G contains a l l the constants and - r — 7 — f o r f + g any non-negative f, g i n G which have d i s j o i n t zero s e t s . Then sG = sup{n : there e x i s t n d i s j o i n t G-zero-sets which are not countably compact} . and dG = sup{n : there e x i s t n d i s j o i n t G-zero-sets which are not compact} . Proof I t i s obvious that - 33 -sG <_ sup{n : there e x i s t n d i s j o i n t G-zero-sets which are not countably compact} . Suppose Z(g^), Z(g2), Z(g n) are n d i s j o i n t G-zero-sets which are not countably compact. We may assume that g p g 2 , g n are non-negative functions i n G. For each i = 1, 2, •••, n, l e t g'- = g 1 g 2 - - g . _ 1 g i + 1 - v g n g ± + H&2'-s±-lH+l--'&n By hypothesis, g|, g£, g^ are functions i n G. Let g = g{+g 2+**- +g n. Then g e G and g _ 1 ( i ) 2 z(%±) f o r e a c h 1 = 1> 2> '"> n « Thus g - 1(D» -1 -1 g (2), g (n) are not countably compact. Hence sG > n. I t follows that sG >_ sup{n : there e x i s t n d i s j o i n t G-zero-sets which are not countably compact} . Consequently, the e q u a l i t y holds. The proof of the second e q u a l i t y i s s i m i l a r . 2.6 D e f i n i t i o n : Let U = {U n} n_^ be a countable open covering of a t o p o l o g i c a l space X and f be a function i n C(X). n I f f(X - U U^) i s i n f i n i t e f o r every n, then we define i = l f U = oo . n If f(X - U U^) i s f i n i t e f o r some n, then we define i = l - 34 -n fU = i n f ( |f(X - U V ± ) | : n = 1, 2, 3, •••} i = l 2.7 Theorem : For every subring G of C(X), we have sG = sup{gU : g e G and U i s a countable open covering of X} . Proof : Let g be a function i n G such that g "*"(r) i s not countably compact f o r r = r ^ , r 2 , •••» r n . Since g ^ ( r ^ ) i s not countably countably compact, hence there e x i s t s a countable open covering of g "^(r^) which k k has no f i n i t e subcover. We can choose U so that a l l the open sets i n U are d i s j o i n t from g ^ ( r ^ ) whenever i ^ k. Let U be the union of U"*", 2 n n -1 U , U and {X - U g ( r ^ ) } . Then U i s a countable open covering i = l of X and gU > n. I t follows that sG <_ sup{gU : g i s i n G and U i s a countable open covering of X} . On the other hand, suppose g i s i n G and U = {U n} n_^ i s a countable open covering of X such that gU >_ m. Then there are m r e a l n numbers r ^ , •••» r m which are contained i n g(X - U U^) f o r every n. -1 -1 -1 Thus, g (r-^), g ( r 2 ) , g ( r m ) are not countably compact. Hence sG > m. This shows that - 35 -sG >_ sup{gU : g i s i n G and U i s a countable open covering of X} . This completes the proof. oo 2.8 Theorem : Suppose V = ^ i s a countable open covering of X where the closure of each i s a countably compact subset of v ^ + ^ . Then sG = sup{gV : g i s i n G}, f o r every subring G of C(X). 00 Proof : Let f be a function i n C(X) and U = "fUn} ^ be a countable open covering of X. n I f r i s a r e a l number belonging to f(X - U U^) f o r every n, i = l then f ^ ( r ) i s not countably compact. Thus f ^"(r) i s not contained i n f o r every i . Therefore, r belongs to f(X - V n) f o r every n. This implies that fU <_ fV. Hence, f o r every subring G of C(X), we have sG = sup{gV:g i s i n G} 2.9 Theorem : Suppose V = *^n n - i ^ s a c o u n t a b l e open covering of a space X where the closure of each i s a countably compact subset of V^ +^ and X - has k connected components f o r every i . Then ( i ) f o r every subring G of C(X), e i t h e r sG = G D or sG <_ k. ( i i ) i f f o r some j , the k components of X - Vj are not countably compact and there e x i s t s h E C(X) such that h(X-Vj)={1,2,••«k}, then there e x i s t s a subring G of C(X) such that sG = k. - 36 -Proof : ( i ) For every f e C(X), since X - has k components f o r each i , hence f(X - V^) i s i n f i n i t e or contains at most k r e a l numbers. Thus fV = oo or fV <_ k. I t follows from Theorem 2.8 that f o r every subring G of C(X), sG = oo or sG < k. ( i i ) Let G be the c o l l e c t i o n of a l l functions f i n C(X) such that f(X - V^) i s f i n i t e f o r some i . I t i s obvious that G i s a subring of C(X) and h e G. For each i = 1, 2, k, h ^"(i) contains a component of X - Vj which i s not countably compact. Thus h "^(1), h "^"(2), h "''(k) are not countably compact. Hence sG >_ k. For each f e G, fV i s f i n i t e . Therefore by ( i ) , fV <_ k. I t follows from Theorem 2.8 that sG < k. Consequently, sG = k. 2.10 C o r o l l a r y : (1) For every subring G of C(R), sG = oo or sG <_ 2. (2) For every subring G of C(R n) where n >_ 2, sG = oo or sG <_ 1. (3) There e x i s t s a subring G of C(R) where sG = 2. (4) There e x i s t s a subring G of C(R n) where sG = 1. Proof : Let V n = (-n;n). Then (1) and (3) follow from Theorem 2.9. Let V m = (-m;m) x (-m;m) x ... x (-m;m) (n terms). Then (2) and (4) follow from Theorem 2.9. 2.11 D e f i n i t i o n : Let g e C(X) and r e R. r i s s a i d to be an s-accumulation point of g i f f f o r every 6 > 0, g ^([r-S;r+6]) i s not - 37 -countably compact, r i s sa i d to be an d-accumulation point of g i f f f o r every 6 > 0, g X([r-<5;r+5 ]) i s not compact. Remark : If g "^(r) i s not countably compact (r e s p e c t i v e l y , compact), then r i s an s-accumulation point ( r e s p e c t i v e l y , an d-accumbation point) of g. But the converse i s not true. Example : Let X = {(x , s i n ^) : x > 0} U {(0,0)} and g be the function defined by : g ( ( x , s i n - ) ) = x and g((0,0)) = 0. Then g e C(X) and g ''"(O) = {(0,0)} i s compact. But g ^([-S;6]) i s not compact f o r every 6 > 0. Therefore 0 i s an d-accumulation point of g and g ^"(0) i s compact. 2.12 Theorem : Let G be a subring of C(X). Suppose G s a t i s f i e s the conditions (1), (3) and (4) i n D e f i n i t i o n 0.4. Then sG = sup{n : g has n s-accumulation points f o r some g i n G} . dG = sup{n : g has n d-accumulation points f o r some g i n G} . Proof : I t follows from.Theorem 2.6 that i sG = sup{n : there e x i s t n d i s j o i n t G-zero-sets which are not countably compact} . - 38 -Let g e G such that g has n s-accumulation points , r 2 , • • • , r n . Let 6 > 0 be such that ;r.+<5] and [r^-5;r.+5] are d i s j o i n t whenever i ^ j . From lemma 0.10, g - 1 ( [ r - f i ; r +6]), g - 1 ( [ r -<S;r +«]),•••,g _ 1([r n-fi;r n+S]) are n d i s j o i n t G-zero-sets which are not countably compact. Thus sG >_ n. This shows that. sG >_ sup{n : g has n s-accumulation points f o r some g i n G}. If g ^ ( r ) i s not countably compact, then r i s an s-accumulation point of g. Hence sG < sup{n : g has n s-accumulation points f o r some g e G} . Consequently, the e q u a l i t y holds. The proof of the second e q u a l i t y i s s i m i l a r . 2.13 Theorem : Let G be a subring of C(X). I f G i s admissible, then sG = sup{n : n = 0 or g 1 ( 1 ) , g 1 ( 2 ) , •••, g 1 ( n ) are not countably compact f o r some g i n G} = sup{n : there e x i s t n d i s j o i n t G-zero-sets which are not countably compact} = sup{n : g has n s-accumulation points f o r some g i n G} . - 39 -and —1 —1 —1 dG = sup{n : n = 0 or g (1), g (2), g (n) are not compact f o r some g i n G} = sup{n : there e x i s t n d i s j o i n t G-zero-sets which are not compact} = sup{n : g has n d-accumulation points f o r some g i n G} . Proof : I t follows from Theorem 2.3, Theorem 2.5, and Theorem 2.12. Section 4 : For every admissible subring of C(X), dG* = dG and sG*=sG. There e x i s t s a n a t u r a l r e l a t i o n between dG and the G*-compactification of X. 2.14 Theorem : Let G be an admissible subring of C(X) and Y be the G*-compactification of X. I f dG i s f i n i t e , then |Y — Xj = dG. Conversely, i f J Y - X] i s f i n i t e , then dG = | Y - x| . Proof : Let g be any function i n G* and g" be the extension of g over Y . I f r i s a d-accumulation point of g, then g ^ ([ r-6 ; r-HS ]) i s not compact f o r every 6 > 0. Thus g ^ ( [r-S; r+<5 ]) meets Y - X for every S > 0. Hence r i s i n the closure of g"(Y - X). Therefore, i f g has n d-accumulation points, then |ci R["g ( Y - X) ] | >_ n. I t follows that |Y—X| >_ n. Consequently, |Y - x| >_ dG. - 40 -Next, suppose y^, y^, y n are n points i n Y - X. Then there e x i s t s a function g" e G * ( Y ) such that "g(y^) = i f o r each i=l , 2,«««,n. _ _ 1 Let g be the r e s t r i c t i o n of g on X. For every <5 > 0 , g ( [ i - < 5 ; i + < 5 ] ) i s a neighborhood of y^ and X i s dense i n Y . Therefore, y. e C l Y [ g _ 1 ( [ i - 6 ; i + 6 ] ) f\ X] = C l ^ g " ^ [ i - S ; i + 5 ] ) ] . Hence g ^"([i-<5; i+S]) i s not compact f o r every § > 0 and every i=l , 2,«««,n. This means that 1 , 2 , n are d-accumulation points of g. Thus n < dG*. Consequently, i f | Y - X| > n, then dG = dG* > n. Suppose | Y - X| i s f i n i t e . Then dG >_ | Y - X|. But i t i s always true that dG <_ | Y - X|. Hence | Y - Xj = dG. Conversely, suppose dG i s f i n i t e . Assume that | Y - X| >_ dG + 1 . Then dG ^ dG + 1 which i s a c o n t r a d i c t i o n . Therefore, |Y - x | _< dG . I t follows that | Y - x | = dG. 2 . 1 5 Theorem : Let G be an admissible subring of C(X). Suppose dG i s f i n i t e . Then X i s l o c a l l y compact. Proof : Let Y be the G*-compactification of X. By Theorem 2 . 1 4 , | Y - x | = dG i s f i n i t e . Therefore X i s l o c a l l y compact being an open subset of the compact space Y . Chapter 3 For purposes of i l l u s t r a t i o n , we study two s p e c i a l subrings of C*(X). Moreover, some algebraic structure of these two subrings t e l l s a l o t about the t o p o l o g i c a l properties of the space X. Section 1 : D(X). 3.1 D e f i n i t i o n : Given a t o p o l o g i c a l space X, l e t D(X) be the c o l l e c t i o n of a l l functions f i n C(X) such that f(A) i s f i n i t e f o r every closed d i s c r e t e subset A of X. 3.2 Theorem : For any t o p o l o g i c a l space X, D(X) i s a subring of C(X). Proof : Let f, g be any two functions i n D(X). For every closed d i s c r e t e subset A of X, f(A) and g(A) are f i n i t e . Thus (f - g)(A) and (fg)(A) are also f i n i t e . Hence f - g e D(X) and fg e D(X). This shows that D(X) i s a subring of C(X). 3.3 Theorem : For any t o p o l o g i c a l space X, D(X) has the following properties : (1) I f g e D(X), then |g| e D(X). (2) I f g e D(X) and Z(g) = ij>, then - e D(X). (3) I f g e D(X) and h e C(R), then h-g e D(X). (4) If D(X) determines the topology of X, then D(X) i s admissible. (5) I f X i s l o c a l l y compact, then D(X) i s admissible. - 42 -Proof : (1) For any closed d i s c r e t e subset A of X, |g|(A) i s f i n i t e since g(A) i s f i n i t e . Hence |g| e D(X). (2) For any closed d i s c r e t e subset A of X, (—) (A) i s f i n i t e since g(A) i s f i n i t e . Hence — e D(X). (3) For any closed d i s c r e t e subset A of X, (h«g)(A) i s f i n i t e since g(A) i s f i n i t e . Hence h«g e D(X). (4) Obviously, D(X) contains a l l the constant functions. Let g e D(X) and r e R. For every closed d i s c r e t e subset A of X, since g(A) i s f i n i t e , hence [(g A r ) V ( - r ) ] ( A ) i s f i n i t e . Therefore, (g A r ) V ( - r ) e D(X) f o r every g E D(X) and r e R. Let f, g be any two non-negative functions i n D(X) such that Z(f) H Z(g) = <f>. Then Z(f + g) = <j>. By (2), f * e D(X) . Hence f e D(X). f + S By D e f i n i t i o n 0.4, i f D(X) determines the topology of X, then D(X) i s admissible. (5) Suppose X i s l o c a l l y compact. Let B be any closed subset of X and x 4 B. Then there e x i s t s a compact neighborhood K of x such that K H B = <J>. Now, X i s completely regular and K i s a neighborhood of x. Therefore, there e x i s t s a function f e C(X) such that f(x) = 1 and X - K £ Z ( f ) . We wish to show that f e D(X). Let A be any closed d i s c r e t e subset of X. Since K i s compact, hence A (\ K i s f i n i t e . Thus f(A H K) i s f i n i t e . I t follows that f(A) = f(A Pl K) U {0}. i s f i n i t e . Therefore, f £ D(X). We also have f(x) = 1 and B X - K Z ( f ) . This - 43 -shows that D(X) determines the topology of X. Hence by ( 4 ) , D(X) i s admissible. 3 . 4 Theorem : Let f be an a r b i t r a r y function i n C(X). Then f e D(X) i f f f i s closed and bounded. Proof : Assume that f e D(X). Suppose f i s unbounded . Then there oo I I e x i s t s a subset A = {x„} , of X such that f(x_) > n and n n=± 1 1 1 1 | f ( x n + ^ ) | > | f ( x n ) | f o r every n. I t i s obvious that A has no accumulation po i n t s . Thus A i s a closed d i s c r e t e subset of X. Hence f(A) i s f i n i t e . This i s impossible. Therefore, f i s bounded. Next, l e t F be any closed subset of X. Suppose f(F) i s not closed. Let r e C l R [ f ( F ) ] - f ( F ) . 00 Then there e x i s t s a subset A = { x n } n = i °^ F such that l i m f C x ^ = r. n-*» I f A has an accumulation point x, then x e F and f(x) = r. But t h i s i s impossible. Therefore, A has no accumulation point. That i s , A i s a closed d i s c r e t e subset of X. Then f(A) i s f i n i t e . This i s impossible since r i f ( F ) , A C. F and l i m f ( x n ) = r. Consequently, f(F) i s closed. n-x» This shows that f i s closed. i Conversely, assume that f i s closed and bounded. Let A be any closed d i s c r e t e subset of X.. For any subset E of f ( A ) , A (\- f "*"(E) i s closed. Thus E = f(A (\ f '''(E)) i s closed. Hence f(A) i s a closed d i s c r e t e subset of R. Furthermore, f(A) i s bounded since f i s bounded. Therefore, f(A) i s f i n i t e . This shows that f e D(X). - 44 -3.5 Theorem : Let S be the c o l l e c t i o n of a l l functions f i n C(X) such that fU i s f i n i t e f o r every countable open covering U of X. Then S = D(X). Proof : Let g e D(X) be given and U = {Un}00 .. be a countable open n=l covering of X. We may assume that U n C . U n + ^ f o r every n. Suppose gU i s i n f i n i t e . Then g(X - U n) i s i n f i n i t e f o r every n. We can f i n d a 00 subset A = { x n ^ n = i °^ ^ s a t i s f y i n g : (1) f o r every n, there e x i s t s n' such that {x^,x 2,•••,x n} Q U n?, { x n + 1,x n + 2,...} C X - U n i and n 1 >_ n. (2) g(x ±) ^ g ( x j ) whenever i f j . Then A i s a closed d i s c r e t e subset of X and g(A) i s i n f i n i t e . This i s a c o n t r a d i c t i o n since g e D(X). Hence gU i s f i n i t e f o r every countable open covering U of X. That i s , g e S. Therefore, D(X) C S. Let f e S be given and A be a closed d i s c r e t e subset of X. Suppose f(A) i s i n f i n i t e . Let E = {a,,}" n be a subset of A such that 1 1 n=± fCa^) 4 f ( a j ) whenever i ^ j . For every n, l e t O n be an open set such CO that E H O n = {a n}. Let U = { O n } n = 1 U {X - E}. Then U i s a countable open covering of X. But fU i s i n f i n i t e . This i s a c o n t r a d i c t i o n since f e S. Thus f(A) i s f i n i t e f o r every closed d i s c r e t e subset A of X. That i s , f E D(X). Hence S C. D(X). Consequently, D(X) = S. - 45 -Section 2 : When D(X) = C(X), then the space X is countably compact. 3.6 Theorem : Let A be any subset of a space X. Then the following are equivalent : (1) A is countably compact. (2) D(A)(X) = C(X). (3) D(A)(X) determines the topology of X and D(A)(X) is closed under uniform convergence. Proof : (1) —> (2) : Since A is countably compact, hence A has no closed i n f i n i t e discrete subsets. Thus D(A) = C(A). Hence D(A)(X) = C(A)(X)= = C(X). (2) —> (3) : Obvious. (3) —> (1) : Suppose A is not countably compact. Then there OO exists a countable i n f i n i t e discrete closed subset T = {a_} .. of A. For 1 1 n=± every n, there exists an open set 0 n such that a n e 0 n and T - { a ^ X - 0 n. Since D(A)(X) determines the topology of X, hence there exists f n e D(A)(X) such that f n ( a n ) = , Z(f n) 3 X - 0 n and 00 0 <_ f n <_ — . Evidently, f n e D(A)(X) for every n. Now, l e t g = J f n . 2 n n=l Since D(A)(X) is closed under uniform convergence, hence g e D(A)(X). 1 T ° ° Therefore, g(T) must be f i n i t e . This i s impossible since g(T) = { ~ J n _ i which is i n f i n i t e . Consequently, A must be countably compact. 3.7 Corollary : For an topological space X, the following are equivalent - 46 -(1) X is countably compact. (2) D(X) = C(X). (3) D(X) determines the topology of X and D(X) is closed under uniform convergence. Section 3 : dD(X) is an upper bound of dG where G ranges through a l l admissible subrings of C(X) such that dG is finite. 3.8 Theorem : Let G be an admissible subring of C(X). If dG = m is finite, then dG<_dD(X). Proof : Let Y be the G*-compactification of X. It follows from Theorem 2.14 that j Y - X| = dG = m. By Theorem 2.13, there exists g e G* such that g (r) is not compact i f f r = l,2,-»«,m. Let "g be the extension of g over Y. Now, CLytg^Cr)] - X ± 0 for each r = 1,2, •••,m and | Y - x| = m. Thus g;(Y - X) = {1,2, • • • ,m} . Therefore, m -1 1 1 m -1 1 1 x - u s (CL - - f ; i + 3)) = Y - U g " " ( ( i - f, i + f) i=l i=l is compact . Let f be the function in C(R) defined by 1 1 n, i f n - - ^ £ r < _ n + -j where n is any integer. f(r) = linear, elsewhere - 4 7 -and l e t h = f*g. For every closed d i s c r e t e subset A of X, since m - 1 1 1 m - 1 1 1 X - U g ( ( i - - j ; i + -j)) i s compact, hence A - u . 8 ( ( i - y i + "j)) i = l i = l m - 1 1 1 i s f i n i t e . Also, h maps i j g ( ( i - —; i + — ) ) to {l,2,«-«,m}. Hence 1=1 h(A) i s f i n i t e . This shows that h e D(X). Now, h e D(X) and h _ 1 ( l ) , h "^(2), h "^(m) are not compact. Therefore, dD(X) >_ m = dG. Section 4 : B(X) 3 . 9 D e f i n i t i o n : Given a t o p o l o g i c a l space X, l e t B(X) be the c o l l e c t i o n of a l l functions f e C*(X) such that p e C l o v [ Z ( f - f g ( p ) ) ] PA f o r every p e 3X. 3 . 1 0 Theorem : For any t o p o l o g i c a l space X, B(X) i s a subring of C(X), Proof : Let f, g be any functions i n B(X) and p be an a r b i t r a r y point i n X. Then (c i 3 x [z ( f - f e ( P ))]) n ( c i 3 x [ z ( g - 8 B(p))]) = c i g x [ z ( f - f e ( P ) ) n z ( g - gP(P))] c c i g x [ z ( f - g - (f - g ) p ( P ) ) ] . Therefore, p e C l g x [ Z ( f - g - (f - g ) 5 ( p ) ) ] f o r every p e gX. Hence f - g e B(X). Also, - 48 -( c i p x [ z ( f - f 6 ( P ) ) ] ) n ( c i p x [ z ( g - gP(P))]) c i B X [ z ( f - f B ( P ) ) n z ( g - g B ( P ) ) ] c c i 3 x [ z ( f g - ( fg ) B (p ) ) ] Therefore p e C l o v [ Z ( f g - ( f g ) B ( p ) ) ] f o r every p e BX. Hence pA fg e B(X). Consequently, B(X) i s a subring of C(X). 3.11 Theorem : Let S be the c o l l e c t i o n of a l l functions f e C(X) such that f(Z) i s compact f o r every zero set Z of X. Then S = B(X). . vi Proof : Let g be any f u n c t i o n i n B(X) and Z be an a r b i t r a r y zero set of X. For every p e C l g x Z , p e C l g x [ Z ( g - g 6 ( p ) ) ] since g e B(X). Thus p e ( C l Z) O (Cl [Z(g - g B ( p ) ) ] ) = C l [Z n Z(g - gP(p))]. Thus Z n Z(g - g^(p)) i <(). Therefore, g g(p) e g(Z). It follows that g P ( C l D V Z ) C g(Z). Hence g(Z) = g ^ ( C l D V Z ) i s compact. This shows that pA pA B(X) C S. On the other hand, l e t f be any function i n S. Suppose f 4 B(X) Then there e x i s t s p e BX such that p 4 C l o v [ Z ( f - f 8 ( p ) ) ] . There e x i s t s a zero set Z of X such that Z n Z(f - f 6 ( p ) ) = <(> and p e C1 D VZ. Now, PA f P ( p ) 4 f(Z) since Z f l Z(f - f 6 ( p ) ) = <|>. Since Z i s a zero set of X L 8X Z and f e S , hence f(Z) i s compact. Thus f 8 ( p ) e f B ( C l 0 ^ Z ) C C l n f B ( Z ) = pX R = f ( Z ) . This i s a c o n t r a d i c t i o n . Therefore S C B(X) - 4 9 -Consequently, B(X) = S. 3.12 Theorem : For any t o p o l o g i c a l space X, B(X) has the following properties : (1) I f g e B(X), then |g| e B(X). (2) I f g E B(X) and Z(g) = ty, then - e B(X). (3) I f g E B(X) and h e C(R), then h-g e B(X). ( 4 ) I f B(X) determines the topology of X, then B(X) i s admissible. Proof : (1) : Let h be the function i n C(R) defined by h(r) = |r| f o r every r e R. Then f o r every g E B(X), |g| = h-g e B(X) by (3). (2) : Let g e B(X) such that Z(g) = ty. Since X i s a zero set::, hence g(X) i s compact. Therefore, there e x i s t s 6 > 0 such that [-6;6] H g(X) = ty. Let h be a function i n C(R) defined by : h(r) = \ l i n e a r , i f -6 < r < 6 1 r elsewhere Then - = h-g i s i n B(X) by (3). 6 (3) : Let g E B(X) and h E C(R). For any zero set Z of X, g(Z) i s compact. Thus (h'-g)(Z) i s compact. By Theorem 3.11, h-g E B(X) ( 4 ) : It i s obvious that B(X) contains a l l the constants. Let g be a function i n B(X) and r be any non-negative r e a l number, f o r every zero set Z, g(Z) i s compact. Therefore [(gAr)V(-r)](Z)= g(Z) H t ~ r ; r ] i s compact. Hence (g A r) V ( - r ) i s i n B(X). - 50 -Let f, g be any two non-negative functions i n B(X) with, d i s j o i n t 1 zero s e t s . Then f + g i s i n B(X) and Z(f + g) = <f>. By (2), j—— i s i n B (X). Thus - r - 7 — i s i n B (X) . f + g 3.13 Theorem : For any t o p o l o g i c a l space X, B(X) i s the l a r g e s t subring of C*(X) s a t i s f y i n g the following conditions : (1) B(X) contains a l l the constants, and (2) M p H B(X) = M*P f l B(X) f o r every p e BX. Proof : Suppose G i s a subring of C*(X) s a t i s f y i n g the conditions (1) and (2). Let g e G. For every p e BX, the function g - g B(p) e M*P f l G. By condition (2), g - g 6(p) e M P f l G. Thus p e C l [Z(g - g B ( p ) ) ] . Therefore, g e B(X). Hence G C B(X). It remains to show that B(X) s a t i s f i e s the condi t i o n (2). For every p e BX, i t i s obvious that Mp fl B(X) C . M*P fl B(X) . Let f e M*P fi B ( X ) . Then f B ( p ) = 0 and p e C l Q V [ Z ( f - f B ( p ) ) ] = C l o v Z ( f ) . pA pA Hence f e MP. Consequently, M*P fl B(X) = M P fl B(X) f o r every p e BX. This completes the proof. Section 5 : We r e c a l l that D(X) i s c l o s e l y r e l a t e d to countable compactness. The same r e l a t i o n s hold between B(X) and pseudocompactness. 3.14 Theorem : The following are equivalent : (1) X i s pseudocompact. (2) B(X) = C(X). - 51 -(3) B(X) determines the topology of X and B(X) i s closed under uniform convergence Proof : (1) — > (2) : Since C(X) = C*(X), hence M*p = M p f o r every i n X. By Theorem 3.9, i t follows that B(X) = C(X). (2) — > (3) : Obvious. (3) — > (1) : Assume that X i s not pseudocompact. Then by 00 lemma 0.22 , X has a C-embedded subset A = t a n ^ n = i which i s d i s c r e t e . oo Let {^n} be a countable c o l l e c t i o n of d i s j o i n t open sets where 0 n contains a n for every n. I t follows from Theorem 3.8 that B(X) i s admissible. For every 1 n, l e t f n be a function i n B(X) such that ( i ) f n ( a n ) = — , ( i i ) Z ( f n ) 00 contains X - 0 n and ( i i i ) 0 <_ f < ^ — . Let g = £ f n . Since B(X) 2 n n=l i s closed under uniform convergence, hence g i s i n B(X). Now, Z(g) i s a zero set d i s j o i n t from A, and A i s C-embedded. By lemma 0.23 , A and Z(g) are completely seperated. Therefore, there e x i s t s a zero set Z such that Z contains A and Z i s d i s j o i n t from Z(g). Then g(Z) 1 oo contains g(A) = {—} .. and 0 i s not i n g(Z). Hence g(Z) i s not 2 n n - i closed. Since Z i s a zero set and g i s i n B(X), hence g(Z) i s compact by Theorem 3.7. This i s a c o n t r a d i c t i o n . Consequently, X must be pseudocompact. Section 6 : For a large c l a s s of t o p o l o g i c a l spaces X, B(X) = D(X). We e x h i b i t a t o p o l o g i c a l space X where B(X) f D(X). - 52 -3.15 Theorem : Let X be a t o p o l o g i c a l space. Suppose every countable closed d i s c r e t e subset of X i s completely seperated from every zero set d i s j o i n t from i t . Then B(X) = D(X). Proof : I t follows from Theorem 3.4 and Theorem 3.11 that D(X) B(X). Let f be a bounded f u n c t i o n which i s not i n D(X). Then there e x i s t s a closed subset A of X such that f(A) i s not closed. Then OO f(A) has an accumulation r such that r i f ( A ) . Let E = {a„} .. be a 1 1 n=l subset of A such that l i m f ( a n ) = r . Since r i f ( A ) , hence E has no n->«> accumulation point. Then E i s a closed d i s c r e t e subset of X. Since f (E) Cl f e ( c i VE) and f B (Cl E) i s compact, hence C l 1 J f ( E ) CZ f B ( C l o v E ) . p X p A K p A Thus r E C l _ f ( E ) C f 8 ( C l o v E ) . Theref ore, there e x i s t s p E Cl„,rE such K p A p A that f B ( p ) = r. Now, E (\ Z(f - r) = <|> and E i s a countable closed d i s c r e t e subset of X. By hypothesis, there e x i s t s a zero set Z of X such that Z D E and Z n Z(f - r) = (j>. Then p i C l . v Z ( f - r) = pA = C l o v Z ( f - f 6 ( p ) ) . Therefore, f i B(X). Hence B(X) <Z. D(X) . P A Consequently, B(X) = D(X). 3.16 C o r o l l a r y : I f every countable closed d i s c r e t e subset of X i s completely seperated from every zero set d i s j o i n t from i t , then X i s pseudocompact i f f X i s countably compact. Proof : I t follows from the f a c t that B(X) = D(X) by Theorem 3.15. We see that i f X i s normal, then B(X) = D(X). - 53 -3.17 Theorem : Let X be a t o p o l o g i c a l space. Suppose D(X) i s admissible and the c o l l e c t i o n of a l l D(X)-zero-sets i s closed under countable i n t e r s e c t i o n s . Then every countable closed d i s c r e t e subset of X i s completely seperated from every zero set d i s j o i n t from i t . 00 Proof : Let A = {a,,} .. be a closed d i s c r e t e subset of X and Z be n n=l a zero set such that Z H A = ty. For every n, l e t f n E D(X) such that (1) f n ( a n ) = ^ (2) Z ( f n ) 3 Z U (A - {a n}) and (3) 0 <_ f n < \ . CO Let f = £ f n . By hypothesis, there e x i s t s g E D(X) such that Z(g)=Z(f). n=l I t i s obvious that Z(g) O Z and Z(g) (\ A = ty. Suppose C l g x Z ( g ) H CI ^ f ty. Let p e C l ^ Z C g ) n C l g x A . Then g£(p) = 0. But g ^ ( C l o v A ) = g(A) since g e D(X) and A i s closed i n X. p X Therefore, g^(p) E g(A). Thus there e x i s t s a point x E A such that g(x) = g^(p) = 0. This i s a c o n t r a d i c t i o n since Z(g) H A = ty. Hence C l ^ Z C g ) H C l g x A = ty. Now, C l g x Z ( g ) and d g x ^ are d i s j o i n t compact subsets of the compact space gX. Thus they are completely seperated i n X. I t follows that A and Z are completely seperated i n X. 3.18 C o r o l l a r y : Suppose D(X) i s admissible and the c o l l e c t i o n of a l l D(X)-zero-sets i s closed under countable i n t e r s e c t i o n . Then B(X) = D(X). Thus X i s pseudocompact i f f X i s countably compact. 3.19 Example : The Tychonoff Plank. - 54 -Let o)1 be the smallest uncountable o r d i n a l . Let W be the ordered space of a l l o r d i n a l s l e s s than w1 and W* be the ordered space of a l l or d i n a l s l e s s than + 1. Let N* = N U {w} be the one-point compactification of the space N of the p o s i t i v e integers. The Tychonoff Plank T i s defined as : T = W* x N* - {(w ,co)} . I t has been proved i n [1] that T i s pseudocompact but not countably compact. Thus by C o r o l l a r y 3.7 and Theorem 3.14, D(T) f C(T) and B(T)=C(T). Hence B(T) f D(T). Section 7 : For i l l u s t r a t i o n , we study some well-known t o p o l o g i c a l spaces. (I) The space N of the p o s i t i v e i n tegers. (1) N i s l o c a l l y compact. By Theorem 3.3, D(N) i s admissible. (2) N i s normal. By Theorem 3.15, D(N) = B(N). (3) Since every subset of N i s closed and d i s c r e t e , hence D(N) = {f e C(N) : f(N) i s f i n i t e } . (4) D(N) seperates d i s j o i n t zero sets of N. By Theorem 1.20, gN i s the D(N)-compactification of N. (5) gN - N i s i n f i n i t e . By Theorem 2.14, dD(N) = a> . (6) N i s a metric space. Therefore, sD(N) = dD(N). (II) The Tychonoff Plank T. (1) T i s l o c a l l y compact. Therefore, D(T) i s admissible. (2) From Example 3.19, D(T) f B(T) and B(T) = C(T). (3) Let G = {f e C(T) : there e x i s t s a and m such that f i s constant on {(cr,n) : a > a and n > m}}. Then G = D(T). - 5 5 -Proof : I t i s obvious that G C D(T). Let f E D(T) be given. I t has been proved i n [ 1 ] that there e x i s t s a such that f i s constant on the t a i l {(cx,n) : a _^ a} f o r each n e N. Now, the edge {co-^ } xN i s a closed d i s c r e t e subset of T and f e D(T). Therefore, f({a) 1} x N) i s f i n i t e . Consequently, there e x i s t s m such that f i s constant on {(a,n) : a >_ a and n >_ m} . Hence f e G. This shows that D(X) G . Hence G = D(T). ( 4 ) I t has been proved i n [ 1 ] that 8T i s the one-point compactification of T. Therefore, 3T i s the D(T)-compactification of T. ( 5 ) By Theorem 2 . 1 4 , dD(T) = | 3 T - T | = 1 and dB(T) = 1 . ( 6 ) I t follows from ( 3 ) that sD(T) = 1 . ( 7 ) sB(T) = 1 . Proof : Let f £ B(T) = C(T). Then there e x i s t s a such that f i s constant on {(a,n) : a >_ a} f o r every n e N. Let r Q be the common l i m i t of f on each {a} * N* f o r a > a. I f f ^ ( r ) i s not countably compact, then f ~*"(r) i s not compact. This can occur only when r = r Q . Hence sB(T) = 1 . ( I l l ) The space R of the r e a l numbers : ( 1 ) R i s l o c a l l y compact. Therefore, D(R) i s admissible. ( 2 ) R i s normal. Therefore, D(R) = B(R). (3) Let G = {f E C(R) : there e x i s t s n such that f i s constant on (-co; -n) and also constant on (n;oo)}. Then G = D(R) . - 56 -Proof : I t i s obvious that G C D ( R ) . Let f E D(R) be given. Suppose f o r each n e N, f i s not constant on (n;oo). Since (n;co) i s connected f o r every n E N, hence f((n;co)) i s i n f i n i t e f o r every n E N. Then there e x i s t s a subset OO A = {a_} n of R such that a ,. > a„ > n f o r every n t N and n n=l n+1 n 7 f(a^) ^ ^ ^ a j ^ whenever i f j . The set A i s a closed d i s c r e t e subset of R, and f(A) i s i n f i n i t e . This i s impossible since f £ D ( R ) . Hence there e x i s t s k E N such that f i s constant on (k;oo). S i m i l a r l y , we can prove that there e x i s t s m e N such that f i s constant on (-co;-m). Let n £ N be such that n >_ m and n >_ k. Then f i s constant on (-oo;-n) and also constant on (n;co). Thus f E G. Hence D(R) C G. Therefore, D(R) = G. (4) Let R** = R U {oo;-co} . We define a topology on R** as follows : R i s an open subset of R**. A l l sets of the form {-oo } U (-oo ; r ] , where r E R, are neighborhoods of the point -co . A l l sets of the form {co } U [r ; c o ) , where r £ R, are neighborhoods of the point oo . I t i s obvious that R** i s a compact space. I t follows from (3) that R** i s the D ( R ) - c o m p a c t i f i c a t i o n of R. (5) |R** - R| = 2. By Theorem 2.14, dD(R) = 2. (6) R i s a metric space. Therefore, sD(R) = dD(R). - 57 -B i b l i o g r a p h y [1] Leonard G i l l m a n and Meyer J e r i s o n . Rings of Continuous F u n c t i o n s , Van N o s t r a n d . (1960). [2] James D u g u n d j i . Topology , A l l y n and Bacon, I n c . , B o s t o n . (1966) .
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Admissible subrings of real-valued continuous functions Choo, Eng-Ung 1971
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Title | Admissible subrings of real-valued continuous functions |
Creator |
Choo, Eng-Ung |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | The object of this thesis is to study the relations between the algebraic properties of admissible subrings [Definition 0.4] of the ring C(X) of all real-valued continuous functions on X and the topological properties of the space X. Given an admissible subring G of C(X), there exists a unique G*-compactification [Definition 1.7 and 0.17] Y of X, with the properties that X is G*-embedded [Definition 0.16] in Y and G*(Y) [Definition 0.12] is admissible. If the cardinality |Y - X| of Y - X is finite or dG [Definition 2.1] is finite, then |Y — X| = dG. From Y, we can get the unique G-realcompactification [Definition 1.8] Z of X where X is G-embedded in Z, G(Z) is admissible and Z is G(Z)-realcompact [Definition 1.1]. It is proved that a G-realcompact space X and an H-realcompact space Y are homeomorphic if G and H are isomorphic admissible subrings. Let D(X) be the subring of all closed bounded functions in C(X). Then X is countably compact iff D(X) is admissible and closed under uniform convergence. For any admissible subring G of C(X), if dG is finite, then dG ≤ dD(X). Let B(X) be the subring of all bounded functions in C(X) which niap zero sets to closed sets in R. Then X is pseudocompact iff B(X) is admissible and closed under uniform convergence. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-30 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080480 |
URI | http://hdl.handle.net/2429/33104 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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