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On certain rings of e-valued continuous functions Chew, Kim-Peu 1969

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ON CERTAIN RINGS OP E-VALUED CONTINUOUS FUNCTIONS by KIM-PEU CHEW B.Sc. Nanyang University, Singapore 1964 M.A. University of B r i t i s h Columbia 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT 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 A p r i l , 1969 In 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 and Study. I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s thes,is f o r f i n a n c i a l gain s h a l l not 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 . Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Supervisor: Dr. J . V. Whittaker i i . ABSTRACT Let C(X,E) denote the set of a l l continuous functions from a topological space X into a topological space E. R. Engelking and S. Mr6wka [2] proved that for any E-completely regular space X [Definition 1 .1] , there exists a unique E-compactification v £X [Definitions 2.1 and 3 .1] with the property that every function f i n C(X,E) has an extension f i n C(v EX,E). • It 3.s proved that i f E i s a (*)-topological d i v i s i o n ring•[Definition 5-53 and X i s an E-completely regular space, i then VgX i s the same as the space of a l l E-homomorphisms [Definition 5 . 3 ] from C(X,E) into E. Also, we establish that i f E i s an H-topological ring [Definition 6.1] and X, Y are E-compact spaces [Definition 2 . 1 ] , then X and Y are homeomorphic i f , and only i f , the rings C(X,E) and C(Y,E) are E-isomorphic [Definition 5»3]» Moreover, i f t i s an E-isomorphism from C(X,E) onto C(Y,E) then "t(w) i s the unique homeomorphisms from Y onto X with the property that t ( f ) = f • t(ir) for a l l f i n C(X,E), where ir i s the identity mapping on X and "t is a certain mapping induced by t. In particular, the development of the theory of C(X,E) gives a unified treatment for the cases when E i s the space of a l l re a l numbers or the space of a l l integers. Finally, for a topological ring E, the bounded subring C*(X,E) of C(X,E) is studied. A function f i n C(X,E) belongs to C*(X,E) i f for any O-neighborhood U i n E, there exists i i i . a 0-neighborhood V in E such that f[X]«V c U and V«f[X] c U. The analogous results for C (X,E) follow closely the theory of C(X,E); namely, for any E -completely regular space X [Definition 9 * 5 ] , there exists an E - compactification V^X of X such that every function f i n C (X,E) has an extension f in C (y EX,E); when E i s the space of a l l nationals, r e a l numbers, complex numbers, or the real quaternions, v £X i s j'ust the space of a l l E-homomorphisms from C (X,E) into E. This i s also v a l i d for a topological ring E which s a t i s f i e s certain conditions. Also, two E -compact spaces [Definition 10.1] X and Y are homeomorphic -ft i f , and only i f , the rings C (X,E) and C (Y,E) are E-isomorphic, where E i s any H -topological ring [Definition 1 2 . 8 ] . i v . TABLE OP CONTENTS Page INTRODUCTION 1 CHAPTER 0 PRELIMINARIES 4 CHAPTER I E-COMPLETELY REGULAR SPACES AND E-COMPACT SPACES 7 1. E-Completely Regular Spaces 7 2. E-Compact Spaces 16 3. The Existence of the Maximal E-Compactifi-cation VgX of an E-Completely Regular Space X „ 21 4. Induced Mapping 26 CHAPTER I I RINGS OF E-VALUED CONTINUOUS FUNCTIONS 27 5. Some Models for VgX 27 6. Representation Theorem of E-Homomorphisms and i t s Application 33 7. Construction of the Homeomorphism From Y Onto X Determined "by an E-Isomorphism From -C(X,E) Onto C(Y,E) 38 CHAPTER I I I RINGS OF BOUNDED E-VALUED CONTINUOUS FUNCTIONS. . i f 2 8. Bounded Subsets of a Topological Ring....... 42 9. E -Completely Regular Spaces 45 V. Page •* * 10. E -Compact Spaces and E -Compactlfications.. -* of an E -Completely Regular Space 52 11. Embedding VgX as a Subspace of v^X 6 l 12. Characterization of the Space X by i t s Function Ring C (X,E) 6j> BIBLIOGRAPHY 73 v i . ACKNOWLEDGEMENTS I wish to express my sincere thanks to my supervisor Dr. J . V. Whittaker for his generous and valuable assistance during the preparation of this thesis, and to Miss Doreen Mah for her excellent typing of the thesis. The fi n a n c i a l support of the Canada National Research Council and the University of B r i t i s h Columbia i s gratefully acknowledged. \ \ 1. INTRODUCTION Let C(X,E) denote some kind of algebraic system of a l l continuous functions from a topological space X into a topological space E. During the past twenty years extensive work has been done on C(X, R), where R i s the space of a l l r e a l numbers. In i960, Gillman and Jerison [3] gave a systematic study of the ring C(X,R) on an arbitrary topological space X. They are concerned with the relations between the algebraic properties of C(X,R) and the topological properties of the space X. The ring C(X,Z) (where Z i s the space-of a l l integers) of a l l integer-valued continuous functions on a topological space X has been studied by R. S. Pierce [13] and S. Mrowka [10]. Their works pa r a l l e l l e d known results i n the theory of real-valued continuous functions. Mr6wka, for example, showed that [10, theorem 2 and theorem 3] (a) Every non-zero (ring -) homomorphism cp from C(X,Z) into Z can be written i n the form (*) cp(f) = f(p) for every f i n C(X, Z), where p. Is a fixed point of X, i f and only I f X i s Z-compact. (b) I f X and Y are Z-compact spaces and the rings C(X,Z) and C(Y,Z) are isomorphic, then the spaces X and Y are homeomorphic. The object of this thesis i s to study the relations between the algebraic properties of the function ring C(X,E), where E Is a topological ring, and the topological structures of the spaces X and E. In particular, we give a uniform treatment for the cases when E = R, Z, the space of a l l rational numbers, 2. the space of a l l complex numbers and the space of a l l real quaternions. In Chapter 0, we quote some well-known results which w i l l often be used i n the thesis. In Chapter I, we give a survey of the properties of the class of a l l E-completely regular spaces and the class of a l l E-compact spaces [2,4 and 9 ] . In Chapter I I , we deal with the ring C(X,E), where E i s a topological ring. R. Engelking and S. Mr6wka [2] proved that for any E-completely regular space X, there exists a unique E-compactification VgX of . X such that every function f i n C(X,E) has an extension 7 i n C(v EX,E). It i s proved that i f E i s a (*)-topological di v i s i o n ring [Definition 5.5] and X is an E-completely regular space, then v^X i s the same as the space of a l l E-homomorphisms [Definition 5.3] from C(X,E) into E. Also, we establish that [Theorem 6.7], i f E i s an H-topological ring .[Definition 6.1], and X, Y are E-compact spaces [Definition 2.1], then the rings C(X,E) and C(Y,E) are E-isomorphi [Definition 5*3], i f , and only i f X, Y .are homeomorphic. This implies the well known result [3, Theorem 8 .3] : Two rea l compact spaces X, Y are homeomorphic i f and only i f C(X,R) and C(Y,R) are isomorphic; and [10, Theorem 3 ] : Two Z-compact spaces X,Y are homeomorphic i f and only i f the rings C(X,Z) and C(Y, Z) are isomorphic. We also establish that [Theorem 7.2] i f E i s an H-topological ring, X, Y are E-compact spaces, and t i s an E-isomorphism from C(X,E) onto C(Y,E), then t(-rr) i s the . unique homeomorphism from Y onto X such that t ( f ) = f o ¥(ir) for a l l f i n C(X,E), where ir i s the identity mapping on X, and T is a certain mapping induced by t . This generalizes the result of L. E. Pursell [14, Theorem 2.1]. In Chapter I I I , we consider the bounded subring C (X,E) of C(X,E), where E i s a topological ring. We say that a function f i n C(X,E) belongs to C (X,E) i f for any zero-neighborhood U i n E, there exists a zero-neighborhood V i n E such that f[X]«V c U and V-f[X] c U. We obtain analogous results for C (X,E) which closely follow the theory of C(X,E); namely, for any E -completely regular space X [Definition 9 . 5 ] , there exists an E -compactification VgX of X such that every function f i n C (X,E) has an extension f i n C (v EX,E); when E Is the space of a l l rationals, real numbers, complex numbers or the real quaternions, v gX i s just the space of a l l E-homomorphism from C (X,E) into E. This i s also v a l i d for a topological ring which s a t i s f i e s certain conditions. Also, two E -compact spaces [Definition 10.1] X and Y are homeomorphic i f , and only i f the rings C (X,E) and C (Y,E) are E-isomorphic, where E i s any H -topological ring [Definition 12 .8] . 4. CHAPTER 0 PRELIMINARIES Unless e x p l i c i t l y stated, a l l topological spaces i n consideration are assumed to Toe Hausdorff„ Thus, the abbre-viation "space" always means "Hausdorff topological space',' Therefore, when we "construct" a space, we must check that i t is a Hausdorff space. We assume also a basic knowledge of a general topology and abstract algebra. In this chapter, we set forth some conventions i n notation and terminology, and record some preliminary results. One should refer to [J] and [6] for those undefined terminologies C(X,Y) w i l l denote the set of a l l continuous functions from the space X into the space Y. For each y i n Y, we sha l l denote by j the constant function ^(x) = y for every x i n X; and Y = (y_: y e Y]. 0.1 Definition. We say that a subset § of C(X,Y) determines the topology of X i f {cp [G]: G i s an open subset i n Y and cp e $} i s a subbase for the topology of X. 0.2 Lemma. [3, p. 42] (a) Let $ be a family of mappings from a space X into a space Y that determines the topology of X. A mapping a from a space S into X i s continuous i f and only i f the composite function cp » o" i s continuous for every cp in $. (b) A mapping a from a space into a product X = ) ( a X a i s continuous i f and only i f TT » a i s continuous for each projection tra. 0.3 Definitions. Suppose that F i s a family of functions such that- each member f of F i s on a topological space X into a space Y f. Then there i s a natural mapping a from X into the product X ^ f : f e F^ which i s defined by mapping a point x of X into a member of the product whose f-coordinate i s f ( x ) , i.e. a ( x ) f = f(x) for each f i n F. We sh a l l c a l l a the evaluation map. We say that F distinguishes (or separates) points of X i f for each pair of dis t i n c t points x and y of X, there i s f i n F such that f(x) / f ( y ) . The family F distinguishes (or separates) points and closed sets of X i f for each closed set A of X and each point x of X ~ A there i s f i n F such that f(x) 4 c l f [ A ] . 0.4 Remark. Since the space X i s assumed to be Hausdorff, F separates points and closed sets of X implies F separates points of X. 0.5 Lemma. [6, p. 116] Let F be a family of continuous functions, each member f being on a topological space X to a topological space Y^ .. Then: (a) The evaluation map a i s a continuous function on -X to the product space ^({Y^: f e F}. 6. (b) The function a i s an open map of X onto a[X] i f P distinguishes points and closed sets of X. (c) The function a i s one to one i f and only i f P dis t -inguishes points of X. By virtue of Lemma 0.5 and Remark 0.4, we have: 0.6 Corollary. Let P be the family of functions given i n Lemma 0 .5. Then the evaluation map a i s a homeomorphism from X onto cr[Xj i f P distinguishes points and closed sets of X. ' 0.7 Lemma. [3, p. 92] I f cp i s a continuous function from a space S into a space Y whose r e s t r i c t i o n cpj^ to a dense subset X i s a homeomorphism, then <p[S ~ X] c Y ~ cp[X]. 0.8 Lemma. [J>} p. 5] Let X be a dense subset of the Hausdorff spaces S and T. I f the identity mapping on X has contin-uous extensions a from S into T and t from T into S, 'then a i s a homeomorphism from S onto T and = t. 7. CHAPTER I E-COMPLETELY REGULAR SPACES AND E-COMPACT SPACES §1. E-Completely Regular Spaces. 1.1 Definition. Let E be a space. A space X i s E-completely 00 regular i f U C(X,En) separates points and closed sets of X. n=l 1.2 Proposition. Suppose X i s E-completely regular. Then C(X^E) separates the points of X. Proof. Let x, y be two di s t i n c t points of X. Since X i s Hausdorff and E-completely regular, there i s f i n C(X,En) for some n such that f(x) i c l ( f ( y ) } . Hence f(x) ^ f ( y ) . Thus for some i (1 < i < n ) y (v± » f)(x) ^  (ir± o f ) ( y ) . But ir± • f e C(X,E). This implies that C(X,E) separates the points of X. 1.3 Definition. Let E be a space and X be a subset of a space Y. We say that X i s C(Y,E)-embedded i f every function in C(X,E) can be extended to a function i n C(Y,E). 1.4 Theorem. The following statements are equivalent. (a) X i s E-completely regular. (b) X i s homeomorphic with a C(E ^  3 ;,E)-embedded subspace C(X E ^ of E * 3 ' under the evaluation map. (c) X i s homeomorphic with a subset of E a for some cardinal number a. 8. (d) C(X,E) determines the topology of X. Proof. (a) -* (t>). Let a be the evaluation map from X into E ^ ^ ^ . We shall show that a i s a homeomorphism and a[X] i s C(E C^ X^ E^E)-embedded. By hypothesis, X i s E-completely regular, so C(X,E) separates points of X by Proposition 1.2. By Lemma 0.5, cr i s a one to one continuous map. a i s an open mapping from X onto o[X], For l e t G be a non-empty open subset of X. For each point p in G there i s h i n C(X,En) for some integer n such that h(p) i c l h [X ~ G]. Let h ± ( i = 1,2,:. .n) be the i - t h coordinate function of hj i.e. h^ = 7r\ « h. Then h. e C(X E) ( i = 1,2, ...n). Let IT,, , , > * be the x ^n-^n^, •.') n n) n( y v\ {h-,,hp, ...,h } projection from E > 1 into E 1 1 = E . Then N = cr[X] fl irJZ , J E ~ c l h[X ~ G]] Is an open set i n \ n]_j • * • > a[X] containing the point cr(p), since TT(, \(cr(p)) = ^n x,...,n nj (h 1(p),h 2(p),.. .,h n(p)) = h(p) i clh[X ~ G]. Furthermore, i f q e N then q = g(x) for some x i n X and ir/v, ^ %(q) = r(hlt . . . , h n ) ( t J ( x ^ = ( h l ( x ) > - - - , h n ( x ) ) = h ( * ) ^  clh[X ~ G]. Hence x e G and a(x) - q e a[G]. Therefore a(p) e N c aLG], and a[G] i s open i n cr[X]. This completes the proof that a i s a homeomorphism. Next, for each g i n C(a[X],E), g • a i s i n OO -* (c). The proof i s t r i v i a l . (Let a be the cardinal of C(X,E)). (c) - ( d ) . Since X i s homeomorphic with a subset of E a for some a, we may regard X as a subset of E a. Then the topology of X i s determined by the set § of a l l projections from X into E. But § c C(X,E), hence the topology det-ermined by $ i s contained i n the topology induced by C(X,E), and the l a t t e r i s the smallest topology for X i n which every member of C(X,E) i s continuous. Thus the topology of X is determined by C(X,E). (d) -* (a). Let F be a closed subset of X and p e X ~ P. Since C(X, E) determines the topology of X, there i s a n subbasic open set U = (If. [G. ], where f. e C(X,E) and i= l G^  are open subsets of E, such that p e U and U D P = 0. {f-,,..., f } Let f be the evaluation' map from X into E = E . n n <?-Then f e C(X,E ) by Lemma 0 .5. V = n tt> [G. ] i s an open 1=1 X± 1 set i n E n and f(p) e V. Moreover, V n f[P] = 0} for i f there exists x e P such that f(x) e V, then x e U, and fore, a[X] i s a C(X,E). Thus 7T g»a E C ( E C ( X > E ) , E ) and ^ g o C 7l CT[XJ = S. There-C(E C( X' E),E)-embedded subset of E C ( X > E ) . 10. this contradicts TJ n F = 0. Thus f(p) / c l f [ F ] and f € C(X,E n). Therefore, X i s E-completely regular. The following two corollaries are immediate con-sequences of Theorem 1.4. 1.5 Corollary. Any subspaee of an E-completely regular space i s E-completely regular. 1.6 Corollary. An arbitrary product of E-completely regular spaces i s E-completely regular. 1.7.. Definition. A space X i s O-dimensional i f i t has a base consisting of open and closed sets. 1.8 Corollary. A space X i s O-dimensional i f and only i f X i s E-completely regular for any O-dimensional space E with card E _> 2. Proof. Suppose that X i s O-dimensional and l e t E be a space consisting of more than one point. Suppose F i s closed i n X and p e X ~ F. Since X i s O-dimensional, there exists an open and closed subset U of X such that p e U and U fl P = j2f. Define f: X -» E by f[U] = {a} and f[X ~ U] = {b} where a, b are fixed d i s t i n c t points of E. Then f e C(X,E) and f(p) £ c l f [ F ] . Thus X i s E-completely regular. Conversely, l e t E be a O-dimensional space and X be an E-completely regular space. By Theorem 1.4, C(X,E) determines the topology of X. Let & be a base of E 11. consisting of open and closed sets. Then B = { f ^ f A J : A e a, f e C(X,E)} is a base for X and each element in B i s open and closed. Thus X i s 0-dimensional. Remarks. Urysohn [16] showed that there exists a countable Hausdorff space E such that the only real-valued continuous functions are constant functions. Such a space E i s evidently not completely regular. But i t i s clearly an E-completely regular space, for the identity mapping on E i s i n C(E,E) and i t separates points and closed sets of E. Suppose E-^  i s Eg-completely regular. Then every E-j-completely regular space X i s Eg-completely regular. For l e t F be a closed subset of X and p e X ~ F. Since X i s E 1-completely regular, there exists f i n C(X, E^) for some f i n i t e integer n such that f(p) £ c l f [ F ] . By Corollary 1.6, E^ i s Eg-completely regular, so there exists h in C(E^,E^) for some f i n i t e integer m such that h(f(p)) £ clh [ c l f [ F ] o c l ( h o f ) [ F ] . But h . f e C(X,Ep. Hence X is Eg-completely regular. Thus, we have: (a) I f E-^  i s Eg-completely regular then every E-j-completely regular space i s Eg-completely regular. There are spaces E-^  and Eg such that E-^  i s Eg-completely regular but not conversely. For instance, take 12. = {a,b} with discrete topology and Eg = [0,1] with usual topology. Then i s Eg-completely regular "but Eg i s not E-^-completely regular since C(Eg,E-^) consists only of constant functions. Suppose E i s completely regular. Similar to the proof of (a), every E-completely regular space i s completely regular. But, the class of a l l E-completely regular spaces may "be properly contained i n the class of a l l completely regular spaces. For instance, take E = {a,b} with discrete topology. Then {a,b} is completely regular. The interval [0,1] i s completely regular, but i t i s not E-completetely regular. For the case E = R (the space of a l l real numbers), a space X i s R-completely regular i f and only i f i t i s completely regular, since R i s completely regular and evi-dently, every completely regular space i s R-completely regular. J. de Groot showed that there exists a subset E of the Euclidean plane which contains more than one point and has the property that each continuous function of E into i t s e l f i s either the identity or a constant mapping [2, p. 435]. Consider this set E and l e t x-j_, Xg e E and x1 x 2. Let p = (x 1,x 2) i n E and F = {(x 1,x 1), (Xg,Xg)} ? 2 c E . Then F i s closed in E and p £ F. Suppose there exists a continuous mapping f of E into E such that f(p) £ c l f [ F ] . Then the mapping f restricted to the set A = {(x,x 2): x € E] i s non-constant, otherwise f(p) = f ( x p , x p ) , 13. and this contradicts f(p) £ c l f [ F ] . The function f ! A can be regarded as a function in C(E,E), and since i t i s non-constant, i t i s the identity, i.e. f(x,x 2) = x for every x i n E. In particular, f(p) = f(x-pXg) = x^. Similarly, f | B where B = {(x^,x): x e E] i s non-constant and f J B e C(E,E), thus f(x- L,x) = x for every x i n E, and in particular, f(p) = f ( x 1 , x 2 ) = x^. This leads to a con-tradiction. Therefore, C(E ,E) does not separate points 2 2 and closed sets of E although E is E-completely regular. Thus, i n general, the fact that u C(X,En) separates points . n=l and closed sets of a space X does not imply that C(X,E) separates points and closed sets of X . 1.10 Definition. We say that an operation © on C(X,E) i s defined pointwisely provided that there exists an operation © on E such that ( f © g)(p) = f(p) © g(p) f° r every f, g in C(X,E) and every point p in E. We can extend the above definition for pointwisely defined operations to operations which have more than two arguments as follows. 1.11 Definitions. An algebraic structure E i s a couple (E; {0 o,0 1,...,0 ?,...} ? < a] where the 0^ (?<a) are oper-ations on the set E. The type of E i s the set [n^: §<a] where n- denotes the number of arguments of the operation 0r, 14. An algebraic structure E w i l l be called a topological algebraic structure provided that a Hausdorff topology for E i s given such that a l l operations on E are continuous. Given a space X and a topological algebraic struct-ure E, C(X,E) becomes an algebraic structure of the same type as E i f the operations on C(X,E) are defined point-wisely. Suppose A^ and Ag are two algebraic structures of the same type. Then a homomorphism from A^ into Ag i s a mapping from A-^  into Ag which preserves a l l the operations on A-^ o A one-to-one homomorphism from A^ into Ag Is called an isomorphism. 1.12 Theorem. Let E be a space. For any topological space X there exists an E-completely regular space Y and a con-tinuous map T from X onto Y such that the induced map-ping T1 : g -» g o T i s a one-to-one map from C( Y,E) onto C(X,E). Moreover, i f E i s a topological algebraic structure, then T' preserves a l l pointwisely defined operations, i .e., T' i s an isomorphism from C(Y,E) onto C(X,E). Proof. Define x s. x 1 i n X to mean that f(x) = f(x') for every f e C(X,E). Then = i s an equivalence relation on X. Let Y be the set of a l l equivalence classes of X under =. Define a mapping T from X onto Y as follows: T(X) i s the equivalence class that contains x for each x i n X. 15-With each f i n C ( X , E ) , associate a function Y g in E as follows: g(y) i s the common value of f(x) at every point x of y. Thus f = g o T. Let C* denote the family of a l l such functions g; i.e. g e C» i f and only i f g a T 6 C( X , E ) . NOW, endow Y with the weak topology induced by C. Then C a C ( Y , E ) . The continuity of T follows from Lemma 0.2. Therefore, for any g i n . C ( Y , E ) , g o T i s i n G(X,E) and hence g e C>. Thus C(Y,E) - C 1 and so the topology of Y i s determined by C ( Y 5 E ) . It follows from Theorem 1.4 that Y i s E-completely regular and i t remains to check that Y i s Hausdorff. Given y, y 1 i n Y and y •/ y', there exists x, x 1 e X and x e y, x 1 e y 1. Therefore x ^  x*, this means there i s an f i n C(X,E) such that f(x) ^  f ( x ' ) . Let g be the function i n C associated with f. Then g(y) = f(x) ^  f(x») = g(y') • Since E i s Hausdorff, l e t U,V be disjoint open sets i n E containing g(y) and g(y') respectively. Then g^~[U] and g^~[V] are disjoint open sets i n Y containing y and y* respectively. The induced mapping T1 i s an onto map: for each f in C ( X , E ) , l e t g be the function i n C' associated with f, then r'(g) = g » T = f. T 1 i s one-to-one: l e t g^, g 2 e C(Y,E) and T'(g 1) = T'(g 2) i.e. g-j_ » T = g 2 o T on X. Since T[X] = Y, g 1 = g 2. 16. Suppose E i s a topological algebraic structure. Then C(X,E) and C(Y,E) are algebraic structures of the same type i f the operations on them are defined pointwisely. It can easily be checked that T' i s an isomorphism from C(Y,E) onto C(X,E). 1.13 Remark. Let E be a topological algebraic structure. As a consequence of the foregoing theorem, algebraic properties that hold for a l l C(X,E) with X E-completely regular, hold just as well for a l l C(X,E), with X arbitrary. Perhaps, this i s a reason for studying E-completely regular spaces, for we are interested i n the connections between the algebraic structure of C(X,E) and the topological properties of X. §2. E-Compact Spaces. 2.1 Definition. Let E be a given space. A space X is E-compact i f X i s E-completely regular and there does not exist any space Y which contains X as a proper dense C(Y,E)-embedded subset of Y. . 2.2 Remark. It i s easy to see that E-compactness i s a topological invariant. 2.3 Lemma. An arbitrary closed subset F of E a where a i s any cardinal number, i s E-compact. Proof. Suppose that P i s a closed subset of E a. Then P i s E-completely regular by Corollary 1.5. If p is-not E-compact then there exists a space T which contains P as 17. a proper dense C(TJ)E)-embedded subset. Each projection TT^  from P into E has an extension rrf i n C(T,E). Define a mapping h: p - (TT|(P)) 1 from T into E a, Then h e C(T,Ea) and h| p i s the identity on P. Therefore, h[T] = h[c l T P ] c clh[P] = c l F = F. Thus h i s i n C(T,F). By Lemma 0.7, h[T ~ F] c P ~ h[P] = F ~ P = 0. Hence T = F. This contradicts that F i s a proper subset of T. 2.4 Corollary. Given a space E, then E a i s E-compact for any cardinal number a. Proof. Since E a i s closed i n i t s e l f , by Lemma 2.3, E a i s E-compact. 2.5 Theorem. Given any space X, there exists an E-compact space W and a continuous mapping 8 from X into a dense C(W,E)-embedded subset of W such that the induced mapping 0': h - h • 0 i s one-one from C(W,E) onto C(X,E). Further-more, i f E i s a topological algebraic structure then 61 i s an isomorphism from C(W,E) onto C(X,E) with respect to a l l pointwisely defined operations. Proof. In virtue of Theorem 1.12, there exists an E-completely regular space Y and a continuous mapping T from X onto Y such that the induced mapping T' = g - g « T i s one-one from C(Y,E) onto C(X,E). Since Y i s E-completely regular, by Theorem 1.4 (b), a[Y] i s a C(EC(Y'E^,E)-embedded 18. CYY subset of E ^ 3 1 where a i s the evaluation map from Y into E C ( Y ' E ) . For each h i n C(a[Y],E), g = h « a e C(Y,E) and 7Tg • a = g = h » a. Thus h = '"'"glatY]* Th e ref° r e ? C(CT[Y],E) consists precisely of the restrictions to cr[Y] of ^ a l l the projections r (g e C(Y,E)). Now a function i n o C(a[Y],E) may have many continuous extensions to a l l of E ^ 9 , but a l l of these extensions must agree on a[Y] and hence.also on i t s closure c l a [Y] since E i s Hausdorff. Thus the process of extension provided a one-one mapping, namely, h - 7T h o a | c l a r y j from C(a[Y],E) onto C(cla[Y],E). Since cla[Y] i s a closed subset of the product space C( Y E^ E ^ 3 i t i s E-compact by Lemma 2.3» Let W = cla[Y] and 0 = a • T. Then e[X] = (a o T)[X] = a[Y] which i s dense and C(W,E)-embedded i n ¥. Also the mapping 6 1 : h -» h <> © i s one-one from C(W,E) onto C ( X , E ) . I f E i s a topological algebraic structure then i t i s easy to check that 6' i s an isomorphism from C(W,E) onto C ( X , E ) . Theorem. Given a space E. The following statements are equivalent. (a) X i s E-compact. cfx E) (b) X i s homeomorphic with a C(E ^  ' ;,E)-embedded subset of E C ^ X j E ^ under the evaluation map a, and a[X] 19. i s closed i n E C ( X ' E ) . (c) X i s homeomorphic to a closed subset of E a for some cardinal number a. Proof. (a) - ( b ) . Suppose X i s E-compact. Then X i s E-completely regular, by Theorem 1.4 (b), the evaluation map a: X - E v * ' i s a homeomorphism, and cr[X] i s C(E K •» ;,E)-embedded in E C ( X ' E ) . I f clcr[X] -/ a[X] then cla[X] con-tains a[X] as a proper dense C(cla[X],E)-embedded subset. This means that a[X] i s not E-compact. But a[X] as a homeomorphic image of an E-compact space X' i s E-compact. Thus cla[X] = ff[X], (b) -» (c). The proof i s t r i v i a l . (c) (a). We know that E-compactness i s a topological invariant and by Lemma 2.3, closed subsets i n E a are E-compact, therefore X i s E-compact i f i t i s homeomorphic with some closed subset i n E for some a. The following two corollaries are immediate con-sequences of the above theorem. 2.7 Corollary. Closed subsets of an E-compact space are E-compact. 2.8 Corollary. Arbitrary products of E-compact spaces are again E-compact. 2.9 Remark. Let I = [0,1] and R be the space of a l l real numbers. Then 20. (a) The following are equivalent. ( i ) X i s completely regular. ( i i ) X i s I-completely regular. ( i i i ) X i s R-completely regular. (k) 13, p. l 6 0 ] . A space i s (Hewitt) real-compact i f and only i f i t i s homeomorphic with a closed subspace of a •.product of real l i n e s , (c) [6, p. I l 8 ] , A space i s compact i f and only i f i t i s homeomorphic with a closed subspace of I a for some a. Proof, (a) ( i ) -» ( i i ) . Suppose X i s completely regular. Then C(X,I) separates points and closed sets of X, hence X i s I-completely regular. ( i i ) - ( i i i ) . Since C(X,I n) c c(X,Rn) (n = 1,2, . I-complete regularity of X implies R-complete regularity of X. ( i i i ) - ( i ) . The proof i s given i n Remark 1 .9. In view of the above remark and Theorem 2.6, we have: Proposition, (a) A space X i s compact i f and only i f i t i s I-compact. (b) A space X is (Hewitt) real-compact i f and only i f i t i s R-compact. 21. §3. The Existence of the Maximal E-Compactifica.tLon v £X of an E-Completely Regular Space X. 3.1 Definition. By an E-compactification of X we mean an E-compact space Y which contains X as a dense subset. 3.2 Lemma. Let X be a dense subset of a space T. Then (a) is ^equivalent to (b). (a) X i s C(T,E)-embedded. (b) X i s C(T,Y)-embedded for any E-compact space Y. Proof, (a) - ( b ) . Suppose Y i s E-compact. In view of Theorem 2.6 we may regard Y as a closed subset of some E a. For each g i n C(X,Y), l e t g ± be the i - t h coordinate function of g, i.e. g i = » g ( i < a) where TT± are projections. Then g i e C(X,E). By hypothesis, i t has an extension g| i n C(T,E). Let g* be a function from T into E a whose i - t h coordinate function i s g*. Then g* 6 C(T,Ea) and g* i s an extension of g. In fact g* € C(T,Y), for g*[T] = g*[clX] c clg*[X] = clg[X] c clY = Y. Therefore g* € C(T,Y) i s the extension of g for g i n C(X,Y). (b) - (a). By Lemma 2.4, E i t s e l f i s E-compact. Thus (a) is a special case of (b). 3.3 Theorem. Every E-completely regular space X has an E-compactification v £X such that (*): X i s C(vEX,Y)-embedded for any E-compact space Y. Furthermore, the space 22. VgX i s uniquely determined by X i n the sense that i f an E-compactification T of X has property (*) then there exists a homeomorphism of v^X onto T that leaves X hi pointwise fixed. Proof. Suppose X i s E-completely regular. By Theorem l A , \1o)x the evaluation map a from X Into E * * ; i s a homeo-morphism and o-[X] i s C(EC(X*E),E)-embedded. Identify X and a[X], then X is a C(E°^X*E^Y)-embedded subset C( X E) of E •* for any E-compact space Y (by Lemma 3.2). Therefore X i s C(clX, Y)-embedded for any E-compact space CYX E^ Y. c l X being a closed subset of the E-compact space E ^  * ' is E-compact (by Corollary 2 .7). Take VgX = c l X . Then v X i s an E-compactification of X with property (*). Suppose T i s any E-compactification of X with property (*). The identity mapping on X " has continuous extensions 0 from v £X into T and T from T into v £X. By, Lemma 0.8, 0 i s a homeomorphism from v gX onto T and clearly e(p) = p for every p i n X. Remark. For any E-completely regular space X, l e t SF(X,E) be the collection of a l l E-compactifications of X. Define an order _< on 3(X 3E) as follows: l e t Y, T e 3(X,E). Then Y _< T means that there exists a continuous function f from T into Y such that f | x i s the identity on X. 23. It i s easy to check that (3(X,E), <) i s a p a r t i a l l y ordered set. Furthermore, T < v gX for a l l T in ^(X^E), since the identity map on X has a continuous extension from v EX into T. We s h a l l therefore c a l l VgX the maximal E-compactification of X. 3.5 Remark. The space VgX i s characterized as an E-compact-i f i c a t i o n of X in which X i s C(VgX,E)-embedded. Evidently, the mapping f - f* (where f* i s the extension of f) of C(X,E) onto C(vEX,E) preserves a l l pointwisely defined operations. 3.6 Corollary. Let X be an E-completely regular space. Then cla[X] i s a "model" of VgX where a is the evaluation map from X into E°^ X* E^. Proof. This can be seen i n the proof of Theorem 3.3. 3.7 Corollary. Let S c X where X i s E-completely regular. I f S is C(X,E)-embedded then c l VS = v^S. VgA E Proof. Since c l . , YS i s an E-compactification of S i n v EX which S i s C(cl YS,E)-embedded, c l VS = v„S. 3.8 Theorem. An arbitrary intersection of E-compact subsets of an E-completely regular space i s E-compact. 24. Proof. Let Y a be a family of E-compact subsets of an E-completely regular space Y and l e t X = n aY a' For each a, the identity mapping T from X into Y a has a contin-uous extension from v £X into the E-compact space YQ. As T can have only one continuous extension from v^X into Y, these extensions a l l coincide; hence this common extension T* carries v^X into D Y i.e. into X. By Lemma 0.7, Hi Ct Ct ' T*[V EX ~ X] c X ~ T*[X] = X ~ X = 0. Hence X = v £X. This proves that X i s E-compact. 3-9 Theorem. Let T be a continuous mapping from an E-compact space X into an E-completely regular space" Y. Then the to t a l preimage of each E-compact subset of Y i s again E-compact. Proof. Let P be an E-compact subset of Y, and l e t S = T*~[F]. Because X i s E-compact, the identity map a on S has a continuous extension to a mapping a*: VgS -» X. Also, T| s has a continuous extension ( r i g ) * : VgS - F. Since S i s dense i n VgS, both these extensions are determined by their values on S. Now, T|S = T « a, and therefore, ( T | S ) * = (T • a)* = T ° a*. But by Lemma 0.7, a*[v£S ~ S] c X ~ S so that ( T • a*) [v ES ~ S] c Y ~ T[S] = Y ~ F, whereas ( T | S ) * [ V E S ~ S] C F. Therefore, v^S ~ S = 0 or S = «vES, so S i s E-compact. 25. 3.10 Problem: Let X be an E-completely regular but not an E-compact space. Find some conditions on X or E so that the smallest element in (tf(X,E), _<) exists. 3.11 Theorem. Suppose E^, Eg are two topological spaces such that E-^  i s Eg-compact and Eg i s E^-completely regular. Then v„ X z> X for every E,-completely regular space X. ^1 ^2 ± Proof. Since E^ i s Eg-completely regular, E^-complete regularity of a space X implies Eg-complete regularity of X. Thus, given any E-j-completely regular space X, v E X exists, and since E^ i s Eg-compact, every function f i n C(X,E-^) has a unique extension f* i n C( v £ X,E^). Since Eg i s E-j-completely regular, the Eg-compact space v„ X i s E.,-completely regular, so (v„ X) exists, and Eg 1 E 1 Eg i t i s an E-^ compactification of X such that every function g i n C(v E X,E1) has an extension i n C( v g ( v £ X),E^). But, for any function f e C(X,E^), f = g| x for some g i n -C(v E X,E 1), so f has an extension i n C( v £ ( v g x),E-L). 2 1 2 Hence (v,-, X) = X. Therefore, X c v„ X. E l E2 E l E2 E l If R i s the real numbers and C i s the complex numbers then R being a closed subset of C i s C-compact, p and C i s R-compact since C = R . The class of a l l 26. completely regular spaces are precisely those spaces that are R-completely regular or C-completely regular. By Theorem 3»11, £or any completely regular space X, v RX = VQX. §4. Induced Mapping Let E he a topological space and X, Y he E-completely regular spaces. Suppose T i s a continuous function from X into Y. The mapping T' from C( Y,E) into C(X,E) defined "by T'(g) = g • T (g e C(Y,E)) i s called the induced mapping of T. The following proposition can he proved easily [3, p. l 4 l ] . 4.1 Proposition. (a) T'(_e) = _e for every e e E. (b) T' determines the mapping T uniquely. (c) T! i s onto i f and only i f T i s a homeomorphism whose image i s C(Y,E)-embedded i n Y. (d) Suppose E i s O-dimensional. Then T1 i s one-one (into) i f and only i f T[Xj i s dense i n Y. As a consequence of (c), (d) of Proposition 4.1, we have: 4.2 Corollary. Let E be a O-dimensional space, X an E-compact space and Y an E-completely regular space. Let T: X -> Y be continuous and T f : C(Y,E) - C(X,E) be i t s induced mapping. Then T ! i s a one-one, onto map i f and only i f T i s a homeomorphism of X onto Y. 27. CHAPTER I I RINGS OF E-VALUED CONTINUOUS FUNCTIONS §5. Some Models for VgX. Let E be a topological ring and X be an E-completely regular space. Then the maximal E-compacti-f i c a t i o n v^X exists. The main problem i n this section i s to find some models of VgX. We are able to show that under certain conditions given on the topological ring E, v^X i s just the set of a l l "E-homomorphisms" from C(X,E) into E. 5.1 Definition. A Hausdorff topological space E i s said to be a topological ring i f E i t s e l f i s a ring and both addit-ion and multiplication are continuous functions from E into E. 5.2 Remark. A topological ring E i s a topological algebraic structure. The type of E i s {2,2}. 5.3 Definition. Suppose that E i s a subring of each of the rings E-j^  and Eg. A ring homomorphism from E^ into Eg ~ is said to be an E-homomorphism i f i t s r e s t r i c t i o n to E i s the identity mapping on E. A one-one E-homomorphism i s called an E-isomorphism. Suppose E i s a topological ring and -X an E-completely regular space. Then X i s homeomorphic with 28. cr[X] where a i s the evaluation map from X into P = E ^ > 1 (Theorem 1.4 (b)) and VgX = clp<j[X] (Corollary 3 .6) . For each point x i n X, ax i s the mapping (ax)(f) = f(x) from C(X,E) into E. Moreover, ax i s an E-homomorphism, since (ax)(_e) = _e(x) = e, V e s E and (ax)(f+g) = (f+g)(x) = f(x) + g(x) = (ax)(f) + (ax)(g), (ax)(fg) = (fg)(x) = f(x)g(x) » (ax)(f) • (ax)(g), (f,g e C(X,E)). Denote the set of a l l E-homomorphisms from C(X,E) into E by H(X,E). Then a[X] c H(X,E) and since the elements i n H(X,E) are mappings from C(X,E) into E, H(X,E) C E ° ( X > E ) = P. I f cl pa[X] « H(X E) then H(X,E) i s a model of v X. This raises the question: under what conditions does H(X,E) = clpa[X]? To answer t h i s , f i r s t of a l l , we show the following lemma. Lemma. For any topological space X and any topological ring E,. H(X,E) i s a closed subset of E C ^ X ' E ) . Proof. For any f, g i n C(X,E), l e t A(f,g) = {p s E C ( X > E ) : p(f+g) = p(f) + p(g)}, where p(f) i s the image of f under the map p. A(f,g) ^  0 because A(f,g) z> CT[X]. We sha l l show that A(f,g) i s closed i n E G ( X ' E ) . For any q e E C( X> E) ~ A(f,g), q(f+g) £ q(f) + q(g). Since E i s Hausdorff, there exist disjoint open sets U, V i n E such that q(f+g) e U and q(f)+q(g) e V. Since the operation + i s a continuous function from ExE 29. into E, there exist neighborhoods V^, of q(f), q(g), respectively, such that V"1 + V*2 c V. The set w = TTf.+g^tU] n 7T f <"[V 1] n TTg^fVg] i s a neighborhood of C(X E) q i n E ^ •* , and for each p e W, we have p( f+g) ^ P(f) + P(g) since p(f+g) e U, p(f)+p(g) e v i + v 2 C V and U n V = 0. Therefore ¥ (1 A(f , g ) = 0. Hence A(f, g) i s a closed subset of E C ( X ' E ) . Similarly, for any f, g i n C(X,E), the set M( f,g) = [p e E G ( X > E ) : p(fg) = p(f)p(g)} i s closed i n E C( X> E). Also, for each e i n E, {p e E0^3^ : p(e) = e} i s a C(X E) closed subset of E ^ 3 '. We observe that H(X,E) = ( fl A(f,g) n M(f,g)) n ( n {peE C( X> E ): p(e) = e}). f,geC(X,S) . •• .eeE Therefore H(X,E) i s a closed subset of E C ^ X ' E ) . 5.5 Definition. A topological division ring E with unity 1 i s said to be a (*)-topological division ring i f : (a) there i s a continuous function x - x* from E Into i t s e l f such that xx* + yy* = 0 implies x = y = 0, where 0 i s the zero element i n E. (b) the function x -+ x~ i s continuous for x ^ 0 i n E. (x~^ denote the multiplicative inverse of x in E). 5.6 Examples of (*)-Topological Division Rings. 30. (1) The ring R of a l l re a l numbers with the usual topology. (2) The ring Q of a l l rational numbers with the rel a t i v e topology induced by R. (3) The complex numbers with the usual topology. (4) The real quaternion ring H with topology so that i t 4 i s homeomorphic with the product R . To be precise, 4 we sh a l l usually i d e n t i f y H with R , the elements 1, i , j , k of the basis of H being id e n t i f i e d respect-i v e l y with (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) 4 of the basis of R . In (1) and (2), x* = x. In (3) and (4), x* i s the conjugate of x. ' Suppose E i s a (*)-topological division ring. For each f i n C(X,E), define a function f* by f*(x) = [ f ( x ) ] * for every x i n X. Then f* e C(X,E) since x -* x* i s continuous on E. I f f e C(X,E) and 0 £ f[X] then the function f " 1 — 1 — 1 defined by f" (x) = [ f ( x ) ] ~ for every x in X i s well-defined and since x - x - 1 i s continuous for x ^ 0 i n E, f " 1 e C(X,E). For any f i n C(X,E), denote Z(f) » [xeX: f(x) = 0}. 5.7 Lemma. Suppose X i s any topological space, and E i s a (*)-topological division ring with .unity 1. Then (1) For any f f 2 , . . . , f n i n C(X,E) (n = 1, 2,...), there 31. exist functions g^, &23 • " 3 &n i n G ( X * E ) such that z ( f l S l + ... + fn&n) = Z ( f ± ) . (2) Suppose p i s a non-zero homomorphism from C(X,E) into E. Then for h in C(X,E), p(h) = 0 implies Z(h) j£ 0. Proof. (1) We prove ( l ) by induction. For n = 1, take g^ to be the constant function 1. Let n = k. By the induction hypothesis, we may assume that there are functions h-^ ,.. •^ h- K-_]_ i n C(X,E) k-1 such that Z( f 1 h-. + ... + f, n h. ,) = n Z( f . ) . Let h = f-Lh-L + ... + f K _ 1h i t_^. Since E i s a (*)-topological division ring, h(x)h*(x) + f (x)f*(x) =0 i f and only i f h(x) = f n ( x ) = 0. Thus Z(hh* + t^f*) = Z(h) n Z ( f n ) . Con-n sequently, Z ( f 1 g 1 + ... + f ^ g ^ = H Z ( f i ) where Si = h i h * . S 2 = h2h*3 " 3 Sn-l = V l h * a n d «n = fn* (2) Suppose h e C(X,E) and Z(h) = 0. Then h"'3" e C(X,E) and hh""1" = 1. Hence h cannot belong to any proper ideal of C(X,E). If p i s a non-zero homomorphism from G(X,E) into E then Kerp = [f e C(X,E): p(f) =0} is a proper ideal of' C(X,E). Thus h £ Ker p or p(h) / 0. We shall now answer the question: When does H(X,E) = c l c[X]? 32. Theorem. Suppose E i s a (*)-topological division ring, and X i s any topological space. Then H(X,E) = clpcr[X] C( X E ^  where P = E ^  > and a is the evaluation map from E into E°( X> E). Proof. We have seen that a[X] c H(X,E) and H(X,E) i s a closed subset of P (Lemma 5A) for any space X and any topological ring E. An arbitrary basic neighborhood of a point p of H(X,E) is a set n n {q € H(X,E): q( f ) e N,} k=l K ^ where N-K i s a neighborhood of p( f^ .) i n E, (k = 1, 2,... ,n By Lemma 5.7 ( l ) , there exist g-^ .. .,g n in C(X,E) such that z( s ( f k - p ( f k ) ) % ) = n z ( f - P ( f k ) ) . k = 1 K £_ ^  k = 1 K £ n Let h = E ( f k - P(f- K))g J S' Since p i s an E-homomorphism, p(h) = 0. By (2) of Lemma 5.7, Z(h) ^  0. n Let x e.Z(h) = n Z(f f c - p ( f f c ) ) . Then f k ( x ) = p(f-K) (k = 1,2, ...n). But (ax)(f k) = f j x ) , hence a(x)(f k) = p ( f k ) (k = 1,2,...,n). Therefore, ax i s i n the 33. neighborhood of p. Thus H(X,E) = c l p a [ X ] . 5.9 Theorem. Suppose E i s a (*)-topological division ring with unity, and X i s an E-completely regular space. Then the set H(X,E) of a l l E-homomorphisms from C(X,E) into E i s a model of v„X. Proof. This i s a consequence of Corollary 3.6 and Theorem 5. 5.10 Remark. Theorem 5.8 holds for any topological ring E such that (1), (2) of Lemma 5.7 are va l i d . For E = Z, the ring of a l l integers, ( l ) of Lemma 5.7 holds, for given f ± € C(X,Z) ( i = 1,...,n), take :e n 2s n g. = f. ( i = 1,2, ...,n). Then Z( E f?) = n Z(f. ). By i=l 1 i = l 1 [lO, §5,(v) ], (2) of Lemma 5.7 holds. Therefore, in view of Remark 5.10, we have • 5.11 Corollary. For any Z-completely regular space X, the set H(X,Z) of a l l Z-homomorphisms from C(X,Z) into Z is a model of vzX. §6. Representation Theorem of E-Homomorphisms and i t s Applications, 6.1 Definition. A topological ring E i s said to be an H-topological ring i f H(X,E) = v £X for any E-completely regular space X, i.e. H(X,E) = cl pa[X] where P = E C ( X ' E ) and a is the evaluation map from X into P. 34. We have seen i n Section 5 that the ring . Z and any (*)-topological division rings are H-topological rings. In this section, we sh a l l assume that E i s always an H-topological ring. 6.2 Theorem. Suppose X i s an E-completely regular space. Then X i s E-compact i f and only i f every E-homomorphism 9 from C(X,E) into E i s fixed i n X, i.e., there exists a unique point x i n X such that 6(f) = f(x) for every f i n C(X,E). Proof. By Theorem 2.6, the E-completely regular space X i s E-compact i f and only i f X i s homeomorphic with C ( X P \ a[X] c P = E ^ » ' under the evaluation map a, and c[X] i s closed i n P. And since E i s an H-topological ring, X i s E-compact i f , and only i f , a[X] = H(X,E). But a[X] = H(X,E) means that, given any 9 in H(X,E), there exists a point x in X such that cr(x) = 9. The "point x i n X is uniquely determined by 0, since a i s a homeomorphism. Thus, 0(f) = o(x)(f) = f(x) for every f i n C(X,E). When E = Z, Q or R, i t can be easily checked that every non-zero homomorphism from C(X,E) into E i s an E-homomorphism. Therefore, as special cases of Theorem 6.2, we have the following corollary. 6.3 Corollary. (1) [10, Theorem 2]. Every non-zero homomorphism cp: C(X,Z) into Z can be written i n the form cp(f) = f(p ) for every f i n C(X,Z) where p Q is a fixed point i n X, 35. . i f and only i f X i s N-compact. (Observe that the space N = {1,2,...3n,...} i s homeomorphic with Z, hence N-compact is equivalent to Z-compact.) (2) [1, theorem 1], Every non-zero homomorphism cp: C(X, Q) -> Q, can be written i n the form cp( f) = f ( p Q ) for a l l f in C(X,Q) where p Q i s a fixed point i n X, i f and only i f X i s Q-compact. (Observe that the space N i s closed i n Q, therefore i t i s Q-compact. This implies every N-compact space i s Q-compact). (3) [3, p. 142]. A space X i s real-compact i f and only i f , to each non-zero homomorphism cp from C(X,R) into R, there corresponds a point x i n X such that cp(f) = f(x) for a l l f i n C(X,R). Corollary. Suppose that X i s an E-completely regular space. Then for each 8 in H(X5E) there exists a unique point x i n v £X such that 9(f) = T(x) 'for a l l f i n C(X,E), where f i s the extension of f i n C(v EX,E). Proof. Given 0 i n H(X,E), we can define a mapping ~0 from C( vEX,E) into E by "e(f) = 0(f) for every f i n C(v EX,E). Clearly 1 belongs to H(v EX,E). Since v £X i s E-compact, by Theorem 6.2 there exists a unique point x i n v EX such that "0(f) = f(x) for a l l f i n C( v EX,E). Thus, 0(f) = T(x) for a l l f i n C(X,E). 36. As an application, we sh a l l examine the problem of determining when a given E-homomorphism from C( Y,E) into C(X,E) i s induced by some continuous mapping from X into Y. 6.5 Theorem. Let t be an E-homomorphism from C( Y,E) into C(X,E). I f Y i s E-completely regular, then there exists a unique continuous mapping T from X into v gY such that •fc(g) = g • T for every g i n C( Y,E), where g i s the extension of g i n C(vgY,E). Proof. For each x i n X, the mapping g -» (tg)(x) i s an E-homomorphism from C(Y,E) into E. By Corollary 6.4, there exists a unique point TX i n v^Y such that (tg)(x) = g( TX) for a l l g i n C( Y,E). The mapping T from X into VgY thus defined, evidently s a t i s f i e s tg = g » T for a l l g i n C(Y,E). Since tg i s continuous and C(VgY,E) determines the topology of v^Y, by Lemma 0.2 T i s continuous. The uniqueness of T follows from Proposition 4.1 (b). 6.6 Remark. In Theorem 6.5, i f Y i s E-compact then Y = v £Y. Therefore T i s a continuous mapping, from X into Y such that tg = g o T for a l l g i n C(Y,E). 6.7 Theorem. Suppose that X and Y are E-compact spaces. Then the ring C(X,E) i s E-isomorphic; with the ring C(Y,E) ( i . e . , C(X,E) and C(Y,E) are isomorphic under an E-isomorphism) i f and only i f X and Y are homeomorphic. 37. Proof. Suppose t i s an E-isomorphism from C(Y,E) onto C(X,E). Then there exists continuous functions T from X into Y and a from Y into X such that (tg)(x) = g(TX) for every x i n X and g i n C(Y,E) and (t^~f)(y) = f(ay) for y i n Y and f i n C(X,E). Then for each y in Y, we have: g(y) = ( t < ~ ( t g ) )(y) = (tg)(ay) = g( T(ay)) for every g i n C(Y,E). Since C(Y,E) separates points of Y, (T o a)(y) = y for a l l y i n Y. Similarly, (a»r)(x) = for a l l x i n X. T i s one-one: for i f TX-^  = TX^ for x^, x^ i n X then x^ = a( TX^) = a( TX 2) = x2« T i s onto: given y in Y then a(y) i s i n X and -r(ay) = y. T has a continuous inverse, namely, the mapping a. Hence T i s a homeomorphism from X onto Y. Conversely, i f T i s a homeomorp"hism from X onto Y, then the induced mapping T ! : g -» g o T i s evidently an E-isomorphism from C(Y,E) onto C(X,E). Remark. I f X i s an E-completely regular spade but not- an E-compact space, then X and v £X are not homeomorphic. But C(X,E) i s E-isomorphic with C( v^X,E) under the mapping f -• f where f e C(X,E) and f i s the extension of f i n C(VgX,E). Therefore, the class of a l l E-compact spaces i s a maximal class of spaces for which Theorem 6.7 holds. 38. Remark. In view of Theorem 6.5 and Theorem 6.7, for any E-compact space X, every E-isomorphism from the ring C(X, E) onto i t s e l f i s the induced mapping of a unique homeo-morphism from X onto i t s e l f . The correspondence establishes an anti-isomorphism from the group of a l l E-automorphisms on C(X, E) onto the group of a l l homeomorphisms on X. To see t h i s , suppose t ^ ( i = 1,2) are E-automorphisms on C(X,E) and T\ ( i = 1,2) are homeomorphisms on X, such that t ±(g) = g c T j L (g € C(X,E), i = 1,2). Then ( t 1 o t 2 ) ( g ) = t x ( t 2 g ) = t x ( g o T 2) = (g o T 2) c T l = g • ( T 2 • T-^), (g e C(X,E)). Hence the E-automorphism ^1 ° ^2 corresponds with the homeomorphism T 2 O T^. Construction of the Homeomorphism From Y onto X Determined by an E-Isomorphism From C(X,E) Onto C(Y,E). Let E be an H-topological ring and X, Y be E-compact spaces. Then according to Theorem 6.5 and Theorem 6.7, for any E-isomorphism t from C(X,E) onto C(Y,E) there i s a unique homeomorphism T from Y onto X such that t ( f ) = f o T for f i n C(X,E). In this section, we shall show that T i s the image of the identity mapping on X under a certain isomorphism induced by t. Let a be any cardinal number. For each g i n C(X,E a), define t(g) i n C(Y,Ea) such that the i - t h coordinate function of "t(g) i s t(-rr± « g) where r± i s 39. the projection from E into the i - t h coordinate space E. Hence TT± • "t(g) = t ( i r i • g) = (ir± « g) » T = ir± • (g 0 T) ( i < a). Therefore 7.1 t(g) = g o T (g 6 C(X,E G)) and i t i s easy to see that ~t i s an E a- Isomorphism from C(X,Ea) onto C(Y,E a), and (tg)[Y] = g[X]. Since X and Y are E-completely regular spaces, we can regard them as subspaces of some product E . Let TT be the identity mapping on X, hence TT^  « ir = ir\ i x ( i _< a). • By (7.1), "t(7r) = 7r « T = T. Hence: 7.2 Theorem. Suppose E i s an H-topological ring and X, Y are E-compact spaces. I f t i s an E-isomorphism from C(X,E] onto C(Y,E), then T(7r) i s the unique homeomorphism from Y onto X such that t ( f ) = f « for a l l f i n C(X,E). To end this section, we sh a l l examine when every homomorphism i s an E-homomorphism and give some consequences of Theorem 6.7 and Theorem 7.2. 7.3 Lemma. Suppose E i s a topological ring with a unity 1 and that the zero homomorphism and the identity mapping are the only homomorphisms from E to E . Then every homomorphism t from C(X,E) Into C(Y,E) Is an E-homomorphism. Proof. For each y in Y, the correspondence e -» (te)(y) 40. is a homomorphism from E into E. By hypothesis, i t i s either the zero homomorphism or the identity mapping on E. But since 1 i s the multiplicative unity i n C(X,E), and t is onto, t ( l ) i s the multiplicative unity i n C(Y,E), i.e, t ( l ) = 1. Therefore, 1 - ( t l ) ( y ) = 1. Hence e - (te)(y) i s not the zero homomorphism, so i t i s the Identity mapping on E. We have: (te)(y) = e for a l l e i n E. For e fixed, we have (te)(y) = e for a l l y In Y. Thus te = _e for a l l e i n E, and t i s an E-homomorphism. 7.4 Lemma, (a) [3, Theorem 0.22], The only non-zero homomor-phism of R into I t s e l f i s the identity. (b) The only non-zero homomorphism of Z (the integers) into i t s e l f i s the identity. (c) The only non-zero homomorphism of Q (the rationals) into i t s e l f i s the identity. Proof. Similar to the proof of Theorem 0.-22 i n [3]. The following two theorems are consequences of Theorem 6.7, Lemma 7.3 and Lemma 7.4. 7.5 Theorem. [3, Theorem 8.3], Two real compact spaces X and Y are homeomorphic i f and only i f ' C(X,R) and C(Y,R) are isomorphic. 7.6 Theorem. [10, Theorem 2]. Two Z-compact spaces X and Y are homeomorphic i f and only i f e(X,Z) and C(Y,Z) are isomorphic. 41. 7.7 Theorem. Two Q-compact spaces X and Y are homeomorphic i f and only i f C(X,Q) and C(Y,Q) are isomorphism. 7.8 Lemma [6, Problem J, p. 103]. Any two open convex subsets of the n-Euclidean space R n are homeomorphic. Since R n i t s e l f i s an open convex set, and R n i s an R-compact space (Corollary 2.4), any open convex subset of R n i s an R-compact space. By Theorem 7.2 and Lemmas 7.3, 7.4, we have: 7.9 Theorem [14, Theorem 2.1]. I f X and Y are open convex subsets of R n (n f i n i t e ) and t i s an isomorphism from C(X,R) onto C(Y,R) then "t( ir) i s the unique homeomorphism from Y onto X with t ( f ) = f(t(ir)) for a l l f i n C(X,R), where ir i s the identity mapping on X and ~t i s the iso-morphism from C(X,Rn) onto C(Y,Rn) defined by ir± » T(g) = t(7r i o g) (1 < i < n) for a l l g i n C(X,R n). 42. CHAPTER I I I RINGS OF BOUNDED E-VALUED CONTINUOUS FUNCTIONS §8. Bounded Subsets of a Topological Ring. 8.1 Definition. A subset S of a topological ring E i s right bounded i f for any neighborhood U of 0, there exists a neighborhood V of 0 such that V*S <= U where V»S i s the set {vs : v e V, s e S}. Left-boundedness i s si m i l a r l y defined, and a subset of E i s bounded i f It i s both l e f t and right bounded. 8.2 Proposition. Any discrete topological ring E i s bounded in i t s e l f . Proof. Since {0} is a 0-neighborhood and (0} «E = E*{0} = {0}, which i s contained In any 0-neighborhood, E is bounded i n i t s e l f . 8.3 Proposition, (a) Any subset of a bounded set i n a topological ring E i s bounded. (b) The union of a f i n i t e number of bounded subsets of a topological ring E is bounded. Proof. The proof i s t r i v i a l . 8.4 Proposition. I f S and T are bounded subsets of a topological ring, so are S+T and S«T. Proof. Suppose U i s any O-neighborhood. Since (x, y) x+y i s continuous on E , i n particular at the point (0,0), there exists a O-neighborhood ¥ such that ¥ + ¥ c U. Since S and T are bounded, there exists a O-neighborhood V such that V«S, V«T, S-V and T« V are contained i n ¥. Therefore, V-(S + T) c V«S + V T c ¥ + ¥ c U and (S + T)-V c S-V + T-V c ¥ + ¥ c U. Hence S + T i s bounded i n E. To see that S-T i s right-bounded, l e t U be any O-neighborhood, and V be a O-neighborhood such that V»T c U. Since S i s bounded, there exists a O-neighborhood ¥ such that ¥«S c V. Then, ¥-(S«T) = (¥-S)-T c V-T c U. Hence S-T i s right-bounded. Left-boundedness of S»T can be proved simi l a r l y . 8.5 Proposition. The closure c l S of a bounded subset S i n a topological ring E i s bounded i n E. Proof. Given any O-neighborhood U, l e t V be a 0-neighborhood such that V - V c U. Since S i s bounded and (x,y) -* xy i s continuous at (0,0), there exists a 0-neighborhood ¥ such that ¥«S cz V and ¥•¥ c v. For any y i n c l S, y + ¥ i s a neighborhood of y, hence (y + ¥) ft S -/ 0 i.e., there exists w e ¥ and s s S such that y + w = s. For any p e W, py = ps - pw e ¥*S - ¥•¥ c V - V c U. Hence ¥-cl S c U. This proves that c l S i s right-bounded. Left-boundedness of c l S can be proved similarly. 44. 8.6 Proposition. Any compact subset K of a topological ring E i s bounded i n E. Proof. Given any O-neighborhood U in E and any point x i n the compact set K, since (x,y) - xy i s continuous from ExE into E, we can find neighborhoods V(x) and W(x) of x and 0 respectively such that V(x)»W(x) c U. A f i n i t e number of the V s cover K. Let N be the inter-section of the corresponding ¥ 1s. We have K>N c U. Hence . K i s left-bounded. Similarly, we can show that K is right-bounded. Suppose E i s a topological ring. Then E a becomes a topological ring provided the operations on E a are defined pointwisely. Therefore, a bounded subset of E a can be defined as i n Definition 8.1. 8.7 Proposition. Suppose E i s a topological ring. A subset W of E a i s bounded i n E a i f and only i f TT^ [¥] i s a bounded subset of E for every projection TT\ ( i < a). Proof. Suppose that W i s a bounded subset of E . Given any O-neighborhood U in E, for any fixed projection (k _< a), -rrk [U] is a O-neighborhood in E . We know that there exists a O-neighborhood V c E a such that W«V c T r ^ t U ] . Hence 7T v[W] .77y[V] = TT-JW-V] c w-Jw^iU]] = U.-45. But ^ [ V ] is a O-neighborhood i n E, so rr^iW] i s l e f t -bounded. Right-boundedness of ir-^W] can be proved similarly. Suppose W i s a subset of E such that ir^JW] is bounded i n E for each i _< a. Given any basic 0-n neighborhood fl irl [N. ] in E a where N. are 0-neigh-k=l ak Jk Jk n borhoods in E. Let N = D N - , then N i s a O-neighborhood k=l Jk in E. Since 7r- [¥] (k < n) are bounded i n E, there it exists a O-neighborhood V i n E such that rr• [W]-V c N Jk n n n for k = 1 ,2,. . .,n. Then W- n TT*"[V] c n i&W c n irf"[N. ]. k=l J k k=l J k k=l J k J k Therefore, W i s left-bounded i n E a. Right-boundedness of ¥ can be shown similarly. §9• E -Completely Regular Spaces. 9.1 Definition. Let X be any topological space and E a topological ring. A function f in C(X,E) i s said to be bounded i f f[X] i s bounded i n E. In view of Proposition" 8.4, the set C (X,E) of a l l bounded functions i n C(X,E) is a subring of C(X,E). 9-2 Definition. Let E be a topological ring. A space X i s said to be E-pseudocompact i f C(X,E) = C*(X,E). 46. Since (Proposition 8.6) any compact subset of a topological ring E is bounded in E, and a continuous image of a compact set is compact, we have: 9.3 Proposition. A compact space i s E-pseudocompact for any topological ring E. 9.4 Remark. It i s clear that i f a topological ring E i s bounded i n i t s e l f then any space X i s E-pseudocompact. In particular, by Proposition 8.2 the space Z of a l l integers is a bounded ring. Thus every space X i s Z-pseudocompact. Thus, our definition.of Z-pseudocompactness is different from the "Z-pseudocompactness" defined i n [13]. This i s because boundedness with respect to the norm i n a normed ring implies the boundedness as we have defined i t , but not con-versely. For instance, the space Z is not bounded with respect to the usual norm, but by Proposition 8.2, i t i s bounded in i t s e l f . However, for the space R of rea l numbers the two notions coincide. Therefore R-pseudocompactness coincides with the pseudocompactness as defined i n [3]. In fact, to be a bounded continuous function from a space X into R i s equivalent to being a bounded continuous function from X into R in the usual sense. 9.5 Definition. Let E be a topological ring. A space X i s -X- * n said to be E -completely regular i f u C (X,E ) separates n=l points and closed sets in X. 47. 9.6 Lemma. Let E be a topological ring. Then (a) f e C*(X,Ea) i f and only i f TT± . f € C*(X,E) for every i _< a. (b) Suppose X i s E*-completely regular, then C*(X,E) separates the points of X. Proof, (a) This i s a consequence of Proposition 8.7. (b) Given x ^  y i n X. Since X .is Hausdorff, {y} i s closed. Thus, there i s f in C*(X,En) such that f(x) ^ f ( y ) , and hence there exists some i (1 _< i < n) with (7r± o f)(x) / (TT± • f ) ( y ) . By (a) r± . f e C*(X,E). Therefore C (X,E) separates the points of X. 9.7 Definition. Let E be a topological ring and X a subset of a space Y. We say that X is C (Y,E)-embedded i f every function f in C (X,E) has an extension to a function in C*(Y,E). 9.8 Lemma. Suppose E i s a topological ring and X an E -completely regular space. Let P = E > ' and a*: X - P* be the evaluation map which i s defined by (a*(x)) f = f(x) . Then (a) a* i s a homeomorphism from X onto <x*[X], (b) cr*[X] is bounded in P*. (c) cl p*a*'[X] i s bounded in P*, and o*[X] is C*(clp*a*[X],E)-embedded i n c l p * a * [ X ] . 48. Proof. (a) By Lemma 0.5 (c), a* i s one-one since (Lemma 9.6 (b)) C (X,E) separates the points of X. By Lemma 0.5 (a), a* is continuous. To see that c* i s an open mapping, l e t G be a non-empty open subset of X. For each point p in G, by E -complete regularity of X, there exists some h in C*(X,En) such that h(p) £ clh[X ~ G]. By Lemma 9.6 (a) TT- o h = f e C (X E) for i = 1,2,. .. ,n. Let ~ % n*( y V) 1 J • • • > ^N) N be the projection from E ^A^-' into E = E . Then N = a*[X] n i r f l _ j E n - clh[X ~ G]j is an open set i n a*[X] containing the point cr*(p), since ^ ( f ^ . . . , f n ) ( G ^ p ) ) = ( f i ( P ) > " - , f n ^ p ) ) = h(p) £ clh[X ~ G]. Furthermore, i f q e N then q = a*(x) for some x i n X, and TT/. „ x(q) = TT/. F N(CI*(X)) = (f (x),..., f (x)) VJ-2.'***,n' ^ 1'' n' " = h(x) ^  clh[X ~ G]. Thus x e G, and hence a*(x) = q e o*[G]. Therefore, a*(p) e N c a*[G], so a*[G] i s open in o*[X], This proves that cr* is an open mapping. Consequently, ex* is a homeomorphism. (b) Since rrf ° a*[X]- = f[X] for every f in C (X,E) and f[X] is bounded In E, by Proposition 8.7, o*[X] i s bounded i n P . (c) Since a*iX] i s bounded in P , i t s closure -x-clcr*[X] i s also bounded in P by Proposition 8.5. 49. For each g in C*(a*[X],E), g . a* € C*(X,E). Thus TT » e C*(P* E) and ir J .x.rvl = g. Let h ~ """g«o* I cla*[X]" T n e n n i s ^ n e continuous extension of g from a*[X] to i t s closure cla*[X], and h[cla*[X] ] c clh[cr*[X]] = clg[a*[X]] which being the closure of the bounded set g[a*[X]] i n E is bounded. Thus h e C*(cla*[X],E). This proves (c). 9.9 Theorem. Let E be a topological ring and X a topological space. Then the following statements are equivalent. (a) X is E -completely regular. (b) X i s homeomorphic to a bounded subset of E a for some cardinal number a. (c) C (X,E) determines the topology of X. Proof, (a) -> (b). This i s a consequence of Lemma 9*8 by -x- . taking a = card C (X,E). (b) - (c). By the hypothesis of (b), we may regard X as a bounded subset of E for some a. Hence the topo-logy of X i s induced by a l l the projections TT\ ( i _< a) on X. Since X i s bounded i n E , by Proposition 8.7, ir^ l x e C (X,E) for a l l 1 < a. Hence the topology for X determined by {T±\x: 1 S. al i s contained i n the topology -x-induced by C (X^E), and the l a t t e r i s the smallest topology for X i n which every member of .0 (X,E) is continuous. Thus the topology of X is determined by C (X,E). 50. (c) - (a). Suppose A i s a closed subset of X and y e X ~ A. Since C (X,E) determines the topology of X, •X-there are functions f l 3 f 2 , . . . 3 f i n C (X,E) such that n 4- n 4-y e fl f T [ U . ] and f) f. [U. ] n A = 0 for some open sets 1=1 1 i=l 1 x (1 _< i _< n) i n E. Let f be the evaluation map from X into E n = E n. Since f ± e C (X,F) ( i < n), by Lemma 9.6 (a), f e C*(X,E n). Since irp « f(y) = 1 i n . n . f (y) e U for i < n, f(y) s fl vT[U. ]. But n 7i> [U. ] 1 1 1=1 * i 1 i=l x i 1 fl f[A ] = 0, otherwise there exists a e A such that f .(a) e U i for i < n. Hence a s fl f. [U. ]. This contradicts i=l n 4-n f. [U. ] n A = 0. Therefore f(y) £ c l f [ A ] . This proves i=l 1 that X is E -completely regular. The following two corollaries are consequences of Propositions 8.3 (a), 8.7 and Theorem 9.9. * 9.10 Corollary. Any subspace of an E -completely regular space is E -completely regular. •* 9.11 Corollary. An arbitrary product of E -completely regular * spaces i s E -completely regular. We have seen that i n the study of the ring of continuous functions, from a space X into a topological ring E, there .is no need to deal with spaces that are not 51. E-completely regular. In the same way, the following theorem says that to study the ring of hounded continuous functions from a space X into a topological ring, we need only to deal with spaces that are E -completely regular. 9.12 Theorem. Let E be a topological ring. Given any topolo-g i c a l space X, there exists an E -completely regular space Y and a continuous function T from X onto Y such that the Induced mapping T': g - g • T is an isomor-# -so-phism from the ring C (Y,E) onto the ring C (X,E). Proof. We write x s x> for x, x ! in X whenever f(x) = f(x') for a l l f in C*(X,E). It is easy to see that s i s an equivalence relation. Let Y be the set of a l l equivalence classes. We define a mapping T from X onto Y as follows: TX is the equivalence class that contains x. With each f in C (X,E), associate a function y g i n E as follows: g(y) i s the common value of f(x) at every point x e y. Thus, f = g © T. Let C' denote the family of a l l such functions g, i.e., g e C i f and only i f g » T e C (X,E). Now endow Y with the weak topo-logy induced by C'. By definition, every function in C is continuous on Y, i.e., C! c C(Y,E). The continuity of T now follows from Lemma 0.2 (a). For any g e C, g o r e C*(X,E). Hence g[Y] = (g . T)[X] i s bounded i n E, so' C! c C*(Y,E). For any h in C*(Y,E), since T i s continuous, h o T e C*(X,E). But 52. this says that h £ C. Therefore, C = C*(Y,E), and i t is clear that the mapping g -• g e T Is an isomorphism from C X(Y,E) onto C*(X,S). It i s evident that i f y and y» are dis t i n c t points of Y, then there exists g e C such that g(y) ^  g'(y). Since E i s a Hausdorff space, this implies that Y i s a Hausdorff space. Hence Y i s E -completely regular, by Theorem 9-9 (c). §10. E -Compact Spaces and E -Compactifications of an E -Completely Regular Space. 10.1 Definition. Let E be a topological ring. By an E*-compact space, we mean an E -completely regular space X such that there does not exist any other space Y which contains X as a proper dense C (Y,EJ-embedded subset. 10.2 Remark. (1) E -compactness is a topological invariant. (2) Any compact, E -completely regular space i s -x-E -compact. 10.3 Lemma. An arbitrary closed and bounded subset >F of E a where E is a topological ring, i s E -compact. Proof. By Theorem 9-9, F is E -completely regular. Sup-* A pose F i s not E -compact, and l e t F be a space which contains F as a dense subset such that each f in C (F,E) has an extension f in C (F,E). Since F i s bounded in E a, 7T-[F] is bounded in E for every projection v.. Thus, 53. •X-T r i 6 C (F,E) ( i _< a). Therefore 77^ has an extension _ -x- A _ 7 T i e C (F, E) ( i < a). Define a mapping h: x - {TT±(X) ) 1 < A A a from P into E , Then h is the identity mapping on F A and h[F] = h [ c l A [ F ] ] c c l h[p] = c l F - F. Hence F E E h' e C(F,F) and by Lemma 0,7, h[F ~ F] c F ~ h[F] = 0. Thus A # F = F. This proves that F is E'-compact. 10.4 Theorem. Let X he an E -completely regular space. Then the following statements are equivalent. (a) X i s E -compact. C ( X F I (b) The evaluation map a* from X into E 3 ; i s a homeomorphism, and a*[X J i s a closed and bounded subset of E C * ( X * E ) . (c) X i s homeomorphic with a closed and bounded subset of E for some cardinal number a. Proof. (a) -.(b). Suppose that X Is E -compact. By -x-definition, i t i s E -completely regular. It follows from Lemma 9.8 that X i s homeomorphic to the bounded subset •X- r> ( Y TP j o*[X] of P = E ^ 3 1 under the evaluation map a* from X into P*. Identify X with a*[X]. Suppose a*[X] i s not closed i n P . Again by Lemma 9.8, clp.x.a*[X] contains tt a*[X] as a proper dense C (cipher* [X],E)~embedded subset. This contradicts that X i s EA-compact. Thus a*[X] i s -x-closed and bounded In P . 54. (b) - (c). Take a = card C*(X,E). (c) -» (a). Suppose X i s homeomorphic to some closed and bounded subset F of E a for some a. By Lemma 10.3, F i s E -compact. Therefore X i s E -compact. Proposition 8.3, 8.7 together with Theorem 10.4 yie l d the following corollary. 10.5 Corollary, (a) Any closed subset of an E -comp ac >  space i s E -compact. (b) The union of a f i n i t e number of E*-compact subsets of an E -completely regular space i s E -compact. (c) An arbitrary product of E -compact spaces i s E -compact. 10.6 Definition. Suppose E i s a topological ring. By an E -compactification of a space X, we mean an E -compact space Y which contains X as a dense subset. 10.7 Lemma. Suppose X i s dense i n T and E i s a topological ring. Then (a) and (b) are equivalent: (a) X i s C*(T,E)-embedded. (b) X. is C(T5Y)-embedded for any E -compact space Y. Proof, (a) -* (b). Suppose Y is an E -compact space. Because of Theorem 10.4, we can regard Y as a closed and bounded subset of E a for some q. For each g i n C(X,Y), let g^ be the i - t h coordinate function of g, i.e., o- = TT oe: ( i < a) where v. i s the projection from E a 55. into the i - t h coordinate space E. Since g[X] c Y and Y i s bounded i n E a, g e C*(X,E) by Lemma 9.6 (a). By hypothesis, g ± has an extension gL i n C*(T,E). Let g be the function from T into E a whose i - t h coordinate function i s g ±. Again by Lemma 9.6 (a), g e C*(T,E a), and g[T] = g[clX] c clgTX] = clg[X] c cl Y = Y. Thus, g e C(T,Y), and clearly g i s the extension of g. (b) - (a). For any g i n C*(X,E), g[X] i s a bounded subset i n E. By Proposition 8.5, clg[X] i s closed and bounded i n E and hence c l g [ X ] i s E -compact by Lemma 10.J. Since g e C(X, clg[X]), by hypothesis, i t has an extension g i n C(T, cl g f X j ) . Therefore g e C(T5E) i s an extension of g. 10.8 Theorem. Suppose E i s a topological ring. Then every E -completely regular space X has an E -compactification v-gX with the following property (*): i f Y is any E -compact space then each function f i n C(X,Y) admits an _. . -x- . -x-extension f in C( v-^X, Y). Furthermore, the space v^X i s uniquely determined by X i n the sense that i f an E -compactification T of X sati s f i e s (*), then there exists -x-a homeomorphism from v g X onto T which leaves X pointwise fixed. -x Proof. Suppose X is E -completely regular. Then take -X- . v EX = cl Mo-*[X] where P* = E C ( X^ E) and cr* i s the 56. evaluation map from X Into P*. By Lemma 9°8 and Lemma 10.3, we can Identify X with a*[X]. Then v*X * * -x-i s an E - compactification of X, and X i s C (v EX,E)-* -X- , . embedded i n VgX» By Lemma 10.7, v^X has property (*), •x-Suppose that T i s an E -compactification of X with property (*). Then the identity mapping on X has continuous extensions T from v^X into T and a from E •X- tt T into VgX. By Lemma 0.8, T i s a homeomorphism from v^X onto T. Clearly T| x i s the identity map on X. •x-10.9 Remark. For any E - >-completely regular space X, the space tt -x-\igX is characterized as an E -compactification of X i n -X- -X-which X i s C ( v£X, E) - embedded <, 10.10 Theorem. Let S c X, where X Is E~-completely regular. If S i s C*(X.E)-embedded then c l „vs = -v*S. VE Proof. The set c l „VS being a closed subset of the tt tt tt E -compact space v £X i s E -compact. Therefore, c l ^ S E * i s an E -compactification of S in which S i s C*( cl^ xS,E)-embedded. Thus, c l ^ S = v ES. 10.11 Theorem. An arbitrary intersection of E -compact subspaces •X- "X* of a given E -completely regular space Is E -compact. 57. Proof. Let ( Y a } u be a family of E*-compact subspaces of an E'""-completely regular space Y, and l e t X = n Y . a a For each a, the identity mapping T from X into Y has a continuous extension from v £X into the E -compact space Y a (Theorem 10.8). As T can have only one continuous •x-exicension from v gX into Y, these extensions a l l coincide; hence this common extension T carries v X into fl Y , E a a> i.e., into X. By Lemma 0.7, ~[ vg'x ~ X] c X ~ "^[X] = 0". Hence v„X = X, so X i s E -compact. E -x-10.12 Theorem. Let T be a continuous function from an E -compact space X into an E -completely regular space Y. Then the t o t a l preimage of each E -compact subset of Y i s E -compact. -x-Proof. Let F be an E -compact subset of Y, and l e t S - T [F]. Because X i s E -compact, the identity map -x-c on S has a continuous extension to a mapping a*:, VgS X (Theorem 10.3). Also, T| s has a continuous extension (T | s ) : VgS - F. Since S i s dense i n v £S, both these _ extensions are determined by their values on S. Now, T|S = T o (j, and therefore ( T | G ) * = ( r ° a)* = T « a*. By Lemma 0.7, cr*[ v*S ~ S ] cz X ~ S, so that ( T • a") [ v^S ~ S] c Y ~ T [ s ] = Y ~ F, whereas ( r|s),'r[ vgS ~ S] c F. Therefore, * . -x- <L- r , * v ES ~ S = 0, or v gS = S. Hence S = T [F] i s E -compact. 58. 10.13 Theorem. Suppose that E i s a topological ring. Then given any space X, there exists an E -compact space W, which contains a continuous image of X as a dense, C (W,S)-embedded subset. Moreover, C (W,E) i s isomorphic with C*(X,E). Proof. By Theorem 9.12, there exists an E -completely regular space Y and a continuous function T from X onto Y such that T': g - g ° T i s an isomorphism from C" (Y,, E) onto C X(X,E). Let ¥ = v*Y. Then Y i s a dense c"(W,E)-embedded subset of ¥, and h - h|y i s an isomorphism from C*(W,E) onto C*(Y,E). Therefore, h - h o T i s an iso-morphism from C*(W,E) onto C*(X 5E). We sha l l now consider the case where E i s the space R of a l l real numbers. 10.14 Lemma. A space X Is compact i f , and only i f i t i s R -compact where R i s the topological ring of a l l real numbers. Proof. Suppose X i s a compact (Hausdorff) space. Then X is normal, and hence i t Is R -completely regular. Since a compact subset of a Hausdorff space i s always closed, X cannot be a proper dense subset of any other space. Therefore X i s R -compact. Conversely, suppose X i s R -compact. By Theorem 10.4, we can regard X as a closed and bounded subset of R a for some cardinal number a. Therefore by Proposition 8.7 59. ir±[X] i s bounded i n R for a l l the projections ir± (l < a). But boundedness of a subset i n the topological ring R i s equivalent to the boundedness of the subset with respect to the usual metric on R. Hence, for each i ( i < a), choose a closed interval I ± i n R such that ir±[X] a I± ( i < a). By Tychonoff's product theorem, I. i s compact, i<a and we see that X i s a closed subset of the compact space )( I.. Therefore, X is compact. l<a By Theorem 10.4 and Lemma 10.14, we have: 10.15 Theorem. A subset of R a where a i s any cardinal number is compact i f and only i f i t i s closed and bounded. (Bounded in the topological ring R a i n the sense of Definition 8.1). 10.16 Remark. Since every metric space s a t i s f i e s the f i r s t axiom of countability, by [6, p. 92] R a i s not metrizable for any uncountable cardinal a. Thus, the concept of boundedness with respect to a metric cannot apply to R a for uncountable cardinal a. But for the space R n where n i s a f i n i t e positive integer, we know that being a bounded subset of the topological ring R n i s equivalent to being a bounded subset with respect to the usual metric on the Euclidean n-space Rn. Therefore, we have as a special case the c l a s s i c a l theorem of Heine-Borel-Lebesgue [6, p. 114]: A subset of Euclidean n-space i s compact i f and only i f i t i s closed and bounded. 10.17 Remark. In view of Lemma 10.14, being a compactification of a completely regular space X i s equivalent to being an R -compactification of X, and the bounded continuous functions from X into R coincide with the usual bounded continuous real-valued functions. Thus v^X i s just the Stone-Cech compactification BX of the completely regular space X. Therefore, Theorem 10.8 i s a generalization of the Stone-Cech compactification theorem. 10.18 Corollary. Suppose E i s any topological ring. Then any E -compact space i s E-compact. The converse i s not true. Proof. I t i s an immediate consequence of Theorem 2.6 (c) and Theorem 10.4. The converse i s not true. For instance, take X = E = Ra Then X i s R-compact but not R -compact i.e., R i s (Hewitt) real-compact but not compact. 10.19 Proposition. Suppose E i s a topological r i n g c I f X i s E-compact as well as E-pseudocompact then X i s E -compact. -X-Proof. Since X i s E-pseudocompact, C(X,E) = C (X«,E). Since X i s E-compact, there does not exist any space Y which contains X as a proper dense subset such that every f i n C(X,E) = C (X,E) has an extension i n C(Y,E). Thus X i s E -compact. 10.20 Problem. Is the converse of Proposition 10.19 true? i . e., Is i t true that: X i s E -compact implies X i s E-pseudo-compact? 61. §11. Embedding u-,X as a Subspace of v^X. E _ E ' Suppose E i s a topological ring and X i s an -x-E -completely regular space. Evidently, X i s also an E-completely regular space. Therefore both of the spaces •X-v EX and ^ X exists. But the constructions of VgX and VgX f a i l to emphasize one essential property of these spaces, namely, that v^X can be embedded i n v-X. To derive this result, we observe that for any f in G(X,E), clgf[Xj i s an E-compact space (Lernma 2*3) and since -f e C(Xsv^l)p f has an extension f i n C(v£X,VgE). By Corollary 10.18 NjgX i s E-compact. Thus f [ c l £ f [X] ] i s an E-compact subspace of v^X by Theorem 3.9» By Theorem 3»8, B = n T<_~ [ c l ^ f [X] ] i s an E-compact space. Moreover fsC(X,E) * i t i s an E-compactification of X i n which X i s C(B,E)-embedded. Thus, we have the following theorem: 11.1 Theorem. v^X = n f ^ ~ [ c l ^ f [ X ] ] c v*X, where f£C(X,E) ^ * ~f e C( v*X, v*E) i s the extension of f. v E ' E -x Since v„X i s a subspace of the E -completely E * -ss-regular space VgX, \>-X i s E -completely regular. Therefore, * * v (v„X) = D exists and i t i s an E -compactification of X E E in which X i s C*(D,E)- embedded. Hence v^v^X) = vJC; * E E E Additional insight of the embedding i s provided by the-62. following lemma and theorem. 11.2 Lemma. Suppose X, Y are E -completely regular spaces and 8 i s a continuous function from X into Y. Let 8* be the mapping: g - g . 8 from C*(Y,E) into C*(X,E). Then G• i s onto implies 0 i s a homeomorphism (into). Proof. Suppose S(x-L) = 9(* 2) for some x.±> x g i n X. Then (9«g)(x;L) = g(8x 1) = g( 6x 2) - (9»g)(x2) f o r a l l g i n C (Y,E). But 8' i s onto, so f ( x 1 ) = f ( x g ) for a l l f € C*(X,E). Since X i s E*-completely regular, C^(X,E) separates points of X. Thus x-^  = Xg. Hence & i s one-one. To see that 0 : 0[X] -* X i s continuous, we observe that the basic closed sets of Y are of the form g [P], where F i s closed i n E and g s C (Y, E), since c'(Y,E) determines the topology of Y. Let f = g * 9. Then f € C*(X,E) and f ^ t P ] = G^fg 4"!?] n e t X ] ] , which i s closed i n X . Hence 0 i s continuous. Thus, 0 i s a homeomorphism. By Corollary 3.6, v-gX = cl pa[X] where a i s the evaluation map from X into P = E C^ X^ E^. By Theorem 10.8, -X-v_X = clB«a*[X] where a* i s the evaluation map from X E Jr -X-* C (X E) into P = E v 9 . Let T denote the r e s t r i c t i o n to a[X] of the projection from P onto P . Clearly, T i s a continuous mapping from c[X] onto a*[X] and T » a = a*. The l a t t e r says that T = a* » a^ "" . Since a^", a* are 63. homeomorphisms, so i s T. Since cl p. x.j*[X] i s s'v-compact i t i s E-compact by Corollary 10.18. By Theorem lA (b) and Lemma, 3.2, T has a continuous extension ~r from clpa[X] into c l p * c r*[X3. 11.3 Theorem. ~ i s a homeomorphism from clpa[X] into clp^cr^X]. Proof. Given f i n C (cl pa[X],E), we are going to find g i n C (clp.x.a*[X],E) such that T5 (g) = S 9 T = f» Since r maps c[X] homeomorphically onto a"* [ X ], we can define a function h: a*[X] -» E by h{ Ty) = f(y)„ (y s a[X]). .Then h = f o T i s In C (a*[X] 5E) 5 since T i s continuous and f i s a bounded continuous function. Since a*[X] i s C (clp.x.a'x"[X],E)-embedded, h has an extension g i n C*(cl p*a*[X],E). But h o T s C*( a[X],E), and ff[X] i s C (clpcr[X],E}-embedded, so h « T has an extension (h ° r) in C*(cl pd"X],E). Since h o T = f l f f [ X ] > ^ h 8 = f • S i n c e (g • ~T) I m = h • T, g « "T - (h o T ) " 0 Hence T»(g) = g * "T = t, i.e., T' C X'(C1 pCT[X],E) . By Lemma 11.2, T Is a homeomorphism. [X] i s a mapping from C (clp^.a*[X],E) onto P §12. Characterization of the Space X by i t s Function Ring C'"' (X„E). "X* Suppose E Is a topological ring and X an E -•x-completely regular space. Then VgX = cl p^.a*[X], where 64. cr"x" i s the evaluation map from X into p X = E C ^ E ' . Clearly, for each x e X, CT*{X) i s an E-homomorphism from c"(X 3E) into • E. Denote by H*(X,E) the set of a l l E-homomorphisms from C*(X,E) into E. Then o*[X] c H*(X,E) c P . 12.1 Lemxaa. H~'(X,S) i s closed i n p"'. Proof. For any fixed f and g i n c ' ( X,E), l e t A(f,g) = {p e P*: p(f) + p(g) - p(f+g)}. A(f,g) 0 because' A(f,g) 3 c*[X]. We s h a l l show that A(f,g) i s closed i n P*. Suppose q £ A(f, g). Then q{f) + q(g) q( f+g). Since E Is Hausdorff, there exist disjoint neighborhoods K and W of q(f) + q(g) and q(f+g) respectively. Since (x,y) - x+y i s a continuous mapping from EXE into E, there exist neighborhoods U of q(f) and V of q(g) such that U + V c K. Let TT^  be the projection from P into the i - t h coordinate space E. Then the set rl [U] n TT^  [ V] 0 7r~, [W] i s a neighborhood of q, and It Is disjoint from A(f,g). Indeed, for any p In TT|~[U] n "^""[V] n 7rJ~g[W], p(f) + P(g) e U + V c K and p(f+g) € W. But K n W = so p(f) + p(g) p(f+g) i.e., p i A(f,g). Therefore, •it A(f,g) i s a closed subset of P „ Similarly, we can prove that the set M{f, g) = {p € P : p(f)p(g) = p(fg)} i s closed in P . Also, for each e e E, the set (e) = {p € P : p(_e) = e} i s closed.. f'r-We o b s e r v e t h a t H*(X,E) = ( n { A ( f , g ) n M ( f , g ) : f , g € C X"(X 5E)}) n ( n < ~ e ) ) . eeE -T h e r e f o r e , H (X, E) b e i n g an i n t e r s e c t i o n o f c l o s e d s e t s i s c l o s e d . As a s e q u e l , c l p x . c r * [ X ] c I-f ( X , E ) . I t i s n a t u r a l t o a s k: when does cl p*o*[X] = H*(X5E)? F i r s t , we s h a l l c o n s i d e r some s p e c i a l c a s e s ; namely, when E i s t h e space R o f a l l r e a l numbers, the space Q o f a l l r a t i o n a l s , t h e space C o f a l l complex numbers o r the space o f a l l r e a l q u a t e r n i o n s . I n the f i r s t two c a s e s , -x- _ f o r e ach x i n E, l e t x = x. m the l a s t two c a s e s , -x-f o r e ach x i n E, l e t x denote t h e c o n j u g a t e o f x i n u 11 * \ l / 2 E; and jhq] = (xx ) ' i s t h e u s u a l norm on E . Then •x-x - x i s a homeomorphism from E i n t o i t s e l f . F u r t h e r m o r e , i t c a r r i e s a bounded s u b s e t o f E i n t o a bounded s e t . To see t h i s , l e t B be a bounded s u b s e t i n E and U be any -x- * O-neighborhood i n E. Then the s e t U = {u ; u e U} i s a O-neighborhood. There e x i s t s a O-neighborhood V i n E such t h a t B-V c U* and V«B c U*. Then, we have B**V* U V*• B* C (U*)* = U. Hence B* = {b": b e B} i s bounded, s i n c e V i s a O-neighborhood. T h e r e f o r e , we have t h e f o l l o w i n g lemma. -x-, -x-12.2 Lemma. F o r each f i n C (X,E)-, t h e f u n c t i o n f d e f i n e d by f*(x) = f ( x ) * i s i n C*(X 5E). 66. We s h a l l c a l l f the conjugate inunction of f. 12.3 Lemma. Let U "be any O-neighborhood. Then (E ~ U)" 1 = { b _ 1 e E: b £ U] i s bounded i n E. Proof. Let fx e E: |jx-j| < e} c U. For any b ^ U, |]b|l 2 c« — 1 1 1 Hence j|b~ |j = < We observe that a set which i s bounded with respect to the norm implies i t i s bounded i n the topological ring. Therefore, (E ~ U) i s bounded. 12.4 Theorem. Let E be any one of the following spaces: Q, R, C -x-or the r e a l quaternions. For any E -completely regular space X, cl p*cr*[Xj = H*(X,E). Proof. The basic neighborhood of a point p i n H (X,E) i s of the form n {q S H*(X,E): i|q(f k) - p(f^) || < e} , k=l where € > 0 and f f c e cf(X,E). Since f k - p ( f k ) e c"(X^S), i t s conjugate function ( i \ - P ( f k ) ) * « C*(X 5E). Let h = \ ( f f c - p ( f k ) ) ( f k - p ( i \ ) ) ! jt—1 —- ——— •x-Clearly, h € C (X,E). Since p i s an E-homomorphism, p(h) = 0. Thus h e Ker p. But p is not a zero-homomorphism, •x-so Ker p is a proper ideal of C (X,E). We claim that h[X] n U ^ 0, where U = {b e E: (|b|| < e }. For otherwise, h[X] c E ~ U, and since b -» b" 1 (b ^  0) i s continuous on 67. E, the function L" 1 defined by h 1(x) - ( h { x ) ) - 1 , (x e X) is continuous. By Lemma 12.3, h e C (X,E). Then -1 -1 h *h € Ker p and h «h = which contradicts that Ker p c (T(X,E). Hence h[X] n U ^ 0. There exists x e X such that h(x) e U, i.e... ||h(x)|t = |i \ ( f k ( x ) - p ( f k ) ) ( f k ( x ) - p ( f k ) ) * | | k—1 = i l i f k ( x ) " P ( f k ) ! l 2 < e 2 k=l Thus, !|fk(x) - p ( f k ) f < e % or j!f k(x) - p ( f k ) | l < e for k = 1,2, ...,n. But a*(x)(f k) = f k ( x ) , so !!a*(x)(f k) - p ( f k ) I! < e for k = 1,2,.„.,n. Therefore, a*(x) i s i n the given basic neighborhood of p i n H (X«E). Hence c l p # o * [ X ] = H*(X,E). 12.5 Definition. E i s a normed ring i f (1) E i s a ring. (2) E i s a normed space. 12.6 Theorem. Suppose that E i s a normed division ring with unity 1 and E has the following properties: (a) b - b~ i s continuous for b ^  0 in E. (b) ||a-bS| = !!a||. lib I I , (a,b s E). (observe that (b) - ||l|| = l ) (c) |l y, b 2|| > |jb 2|! j = 1,2, ...,n. ( b . e E). 1=1 1 J Then clp.x.a*[X] = H*(X,E) for any E*-completely regular 68. space X. P r o o f . The b a s i c n e i g h b o r h o o d o f a p o i n t p i n R A { X 5 E ) i s o f t h e form, n (q e K * ( X , E ) : |!q(f ) - p ( f ) || < g } , k = l -^ n o where € > 0 and f, e C ( X , E ) . Denote h = z ( f v - p{ K)) k = l K i L _ C l e a r l y h e C* {X,E). As p i s an E-homomorphism, p ( h ) = 0 so h e K e r p. But p i s n o t a O-homomorphism, so Ker p £ C * ( X , E ) . We c l a i m t h a t h [ X ] fl U / 03 where U = fb e E : lib || < e 2 } . F o r o t h e r w i s e , h [ X ] c E ~ U 0 We — 1 — 1 show t h a t ( E ~ U)~ = f b ~ : b £ U] i s bounded. G i v e n any b £ U, ||b || > e 2 , and 1 = ||l || = l l b b _ 1 j ! = ] | b | l l i b " 1 I t , so | l b _ 1 | i = < • — . T h e r e f o r e ( E ~ U ) " 1 i s bounded w i t h r e s p e c t t o t h e norm, and hence i t i s bounded. D e f i n e a f u n c t i o n h as f o l l o w s : h (x) = h ( x ) , (x e X ) . Then h € C ( X , E ) , and h»h = 1 e Ker p, which c o n t r a d i c t s -t h a t Ker p £ C * ( X , E ) . Hence h[ X ] n U ^ . There e x i s t s n x s X such t h a t h ( x) e U, i . e . , ||h(x) | l = | l z ( f v ( x ) - p ( f v ) ) k = l * K < e 2 . By ( b ) and ( c ) ||f k(x) - p ( f k ) | l 2 = I l ( f k ( x ) - p ( f k ) ) 2 l l < s 2 . Hence \\t (x) - p ( f f c ) [ [ < e, (k = 1,2, . . . , n ) . But o*(x)(f k) = f k ( x ) , so |lo*(x){f k) - p ( f k ) ! l < e for k = 1, 2,... }no Therefore, a*(x) belongs to the basic neighborhood of p i n Ef'(X,E). Hence, ci ?^a*[X] = H""(X,E). 12.7 Remark. I f the condition (b) i s replaced by (b J j : !jan!| = lla | l n for a l l integers n, (a <s E), then Theorem 12.6 s t i l l holds. Because (b s) implies lib" x|| = - | ] ^ y [ f o r b ^ ° » s o (E ~ U)" 1 i s bounded, where U = (b e E: libjl < e 2}. (b s) also implies !|a |j = j|a|| for a l l a i n E. Therefore, the proof given i n 12.6 i s applicable. 12.8 Definition. A topological ring E i s said to be an Ii -topological r i n g i f H (X,E) = clp^a-iXj for any E -completely regular space X c In the rest of this section, E w i l l denote an •x-H -topological ring. 12.9 Theorem. Suppose X i s an E""-completely regular space. -x-Then X i s E -compact i f , and only i f , every E-homomorphism -X-e from C (X^E) Into E can be written i n the,form e(f) = f(x) for every f i n C (X,E), where x i s a unique fixed point in X. Proof. By Theorem 10.4, the space X Is E -compact i f , and only i f , X i s homeomorphic with a*[X] c P = E \ * > under the evaluation map cr*, and c*[X] i s closed and" 70-. -X- -X-bounded m P . And since E i s an H -topological ring, -X- -X-X i s E -compact i f , and only ±t9 a*[X] = H (X,E). But a*[X] = ff(X,E) means that, given any 6 i n H*(X 3E), there exists a point x i n X, such that cr*(x) = 0. The point x i s uniquely determined by e, since a* i s a homeomor-phism. Thus, 0(f) = ( a*x)(f) = f{x) for every f i n c""(X,E). 12.10 Corollary. Suppose that X i s an E~-completely regular space. Then, for each & i n H X(X,E), there exists a unique point x in VgX such that 0(f) = f(x) for a l l f i n C (X,E), where f i s the extension of f i n C {v_,X,E). -X* ™_ Proof. Given 0 i n K (X,E), we can define a mapping 0 from C (vgX,E) into E by 8(f) = 0(f) for every f" i n C^  ( v* X,E). Clearly, "8 belongs to K"( v^X,E). Since v-^ X -X-i s E -compact, by Theorem 12.9, there exists a unique point •v. , -X-x i n v^X such that g(f) = f(x) for a l l f i n C (VgX,E). Thus, 0(f) = f(x) for a l l f i n (T(X 3E). -x-12.11 Theorem. Let t be an E-homomorphism from C (Y,E) into •x- . -x-C (X,E). I f Y i s E -completely regular, then .there exists •x-a unique continuous mapping T from X into v^Y such that _ t(g) = g " T for every g i n C (Y,E), where g i s the •X- -x-extension of g i n C (v Y 5E). Proof. For each x i n X, the-mapping g - (tg)(x) i s an E-homomorphism from CA(Y,E) into E. By Corollary 12.10, there exists a unique point TX i n v RY such that (tg)(x) = g(rx) for a l l g In C'\%E). The mapping T from X into v^Y thus defined, evidently s a t i s f i e s -X-tg = g * T for a l l g in C ( Y 5 E ) . Since tg i s continuous, and C (v EY,E) determines the topology of v^Y, by Lemma 0.2, T i s continuouso The uniqueness of T follows from the fact that C (Y,E) separates the points of Y. 12.12 Remark. In Theorem 12.11, i f Y i s EX-compact, then v Y - Y. Therefore, T i s a continuous mapping from X into Y such that tg = g • T for a l l g i n C*(Y,E). 12.13 Theorem. Suppose that X and Y are E"*-compact spaces. Then the ring C (X,E) i s E-isomorphic with the ring C (Y,E) -X~ -X* ( i . e . , C (X,E) and G (Y,E) are isomorphic under an E-isomorphism) i f , and only i f , X and Y are homeomorphic. Proof. Suppose t i s an S-isomorphism from C*(Y,E) onto • •X* C (X,E). Then, there exist continuous functions T from yX into Y, and a from Y into X such that: (tg)(x) = g(TX) for each x i n X and g i n c"(Y,E); and ( t * ~ f )(y) = f(ay), for each y i n Y and f i n C (X,E). We haver g(y) = ( t < " ( t g ) ) ( y ) = (tg)(ay) = g( T( ay)) for every g i n C (Y,E) and every y in Y. Since C (Y,E) separates the points of Y, (T « a)(y) = y for a l l y i n Y. Similarly, (a o T)(X) = x for a l l x In X. T i s one-one: for i f TX-^  - TX^ for x^, x^ i n X, 'o hen x = ~( TX 1) = a( TX 2 ) = Xg> 72. T i s o n t o : g i v e n y i n Y, t h e n c ( y ) i s i n X and T(ay) = y. T has a c o n t i n u o u s Inverse., namely, th e mapping a. Hence T i s a homeomorphism from X onto Y. C o n v e r s e l y , i f T i s a homeomorphism f r o m X onto Y, t h e n t h e i n d u c e d mapping Ts : g -> g * r i s e v i d e n t l y an E - i s o m o r p h i s m f r o m C'^Y^E) onto ^ ' ( X . E ) . 12.14 Remark. I f X i s an E ^ - c o m p l e t e l y r e g u l a r space, but n o t * - l e -an E -compact space,, then X and v^X a r e n o t homeomorphic. But C A"(X,E) I s E - I s o m o r p h i c w i t h C*( v*X,E) under t h e mapping: f «* f , where f € CA(X«,E) and f i s t h e e x t e n s i o n o f f i n C ( v £ X , E ) . T h e r e f o r e , t h e c l a s s o f a l l E -compact spaces i s a maximal c l a s s o f spaces f o r which Theorem 12.13 h o l d s . 12.15 Remark. Suppose ' X i s E"-compact. Then e v e r y E - i s o -morphism f r o m t h e r i n g C ' (X 5E) onto I t s e l f i s the i n d u c e d mapping o f a u n i q u e homeomorphism f r o m X onto i t s e l f . The c o r r e s p o n d e n c e e s t a b l i s h e s an a n t 1 - i s o m o r p h i s m from the group o f a l l E-automorphism on C (X E ) , onto the group o f a l l homeomorphisms on X. To see t h i s , suppose t ^ ( I = 1,2) a r e E-automorphisms on G ( X , E ) , and ( I - I,2) a r e homeomorphisms on X, such t h a t t ± ( g ) = g © r± ( g s c ' f X ^ ) ) , Then, ( t 1 o t 2 ) ( g ) = t 1 ( t 2 g ) = t ^ g © T 2) = ( g © T 2) © T L = g © ( T P © TT). Hence t-, © t 0 c o r r e s p o n d s w i t h T0 © T-, . 73 o BIBLIOGRAPHY [1] B l e f k o , R o b e r t L. S t r u c t u r e o f Continuous F u n c t i o n s V I I , P r o c . Kon. Ned. Akad. van Wetensch. A 71 438-441 (1968). [2] E n g e I k i n g , R. and S. Mrowka. On E-compact Spaces, B u l l . Acad. P o l o n . S c i . S e r . Math. Astronom. Phys. 6, 429-435 (1958). [3] G i l l m a n , L. and M. J e r i s o n . Rings o f Continuous F u n c t i o n s , Van N o s t r a n d . (i960). [4] H e r r l i c h , H. and J . Van Der S l a t . P r o p e r t i e s Which a r e C l o s e l y R e l a t e d t o Compactness, N e d e l . Akad. van Wetensch. P r o c . S e r . A 70, 524-529 (1967); Indag. Math. 29 , 524- 529 (1967). [5] K a p l a n s k y , I. T o p o l o g i c a l R i n g s , Amer. J . o f Math. V o l . 69, I53-I83 (1947). [6] K e l l e y , J . L. G e n e r a l Topology, Van N o s t r a n d , New York (1957). [7] Mr6wka, S. On u n i v e r s a l s p a c e s , B u l l , Acad. P o l o n . S c i . , C l . I l l , 4, 479-481 (1956). [8] . A p r o p e r t y o f H e w i t t E x t e n s i o n vX o f T o p o l o g i c a l Spaces, B u l l . Acad. P o l o n , S c i . , Ser.-Math., A s t r . e t Phys., 6, 95-96 (1958). [9] . On E-compact Spaces I I , B u l l . Acad. P o l o n . S c i . V o l . XIV No. 11, 597-605 (1966). 74. [10] . S t r u c t u r e s o f Continuous F u n c t i o n s I I I : R i n g s and. L a t t i c e s o f I n t e g e r - v a l u e d C o n t i n u o u s F u n c t i o n s , P r o c e e d i n g s Kon. Wed. Akad. Wetensch. A 68 74-82 ( 1 9 6 5 ) . [11] s and So D. Shore. S t r u c t u r e s o f C o n t i n u o u s F u n c t i o n s IV: On the R e p r e s e n t a t i o n o f R e a l Homomorphisms D e f i n e d on F u n c t i o n - l a t t i c e s , P r o c e e d i n g s Kon. Red. Akad. Wetensch. A 68, 83-91 ( 1 9 6 5 b Indag. Math. 27, 83-91 (1965). [12] , and . S t r u c t u r e s o f Continuous F u n c t i o n V: On Homomorphisms o f S t r u c t u r e s o f C o n t i n u o u s F u n c t i o n W i t h O - d i m e n s i o n a l Compact Domain, P r o c e e d i n g s Kon. Ned. Akad. Wetensch. A 6 8 , 92 - 9 4 (1965); Indag. Math. 27, 92-94 (1965). [13] P i e r c e , R. S. R i n g s o f I n t e g e r - v a l u e d Continuous F u n c t i o n s , T r a i l . Amer. Math. Soc. V o l . 100, 371-394 ( 1 9 6 l ) . [14] P u r s e l l , L.E. R i n g s o f Continuous F u n c t i o n s on Open Convex S u b s e t s o f R n , P r o c . o f Amer. Math. Soc. V o l . 19 No. 3 581-585 ( 1 9 6 8 ) . [15] Shore, S.D. Homomorphisms o f L a t t i c e s o f I n t e g e r - v a l u e d C o n t i n u o u s F u n c t i o n s 5 . P r o c . Kon. Ned. Akad. Wetensch A 68 No. 3 , 533-538 ( 1 9 6 5 ) ; Indag.. Math. 27 No. 3 , 533-538 (1965) [16] Urysohn, P a u l . Uber d i e M a c h t i g k e i t der zusammenhangenden Menger, Math. Ann. Band 94 , 262-295 (19 2 5 ) . 

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