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

Abstract

Let C(X,E) denote the set of all continuous functions from a topological space X into a topological space E. R. Engelking and S. Mrowka [2] proved that for any E-completely regular space X [Definition 1.1], there exists a unique E-compactification [formula omitted] [Definitions 2.1 and 3.1] with the property that every function f in C(X,E) has an extension f in [formula omitted]. It is proved that if E is a (*)-topological division ring [Definition 5-5] and X is an E-completely regular space, then [formula omitted] is the same as the space of all E-homomorphisms [Definition 5.3] from C(X,E) into E. Also, we establish that if E is an H-topological ring [Definition 6.1] and X, Y are E-compact spaces [Definition 2.1], then X and Y are homeomorphic if, and only if, the rings C(X,E) and C(Y,E) are E-isomorphic [Definition 5.3]. Moreover, if t is an E-isomorphism from C(X,E) onto C(Y,E) then [formula omitted] is the unique homeomorphisms from Y onto X with the property that [formula omitted] for all f in C(X,E), where π is 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 is the space of all real numbers or the space of all integers. Finally, for a topological ring E, the bounded subring C*(X,E) of C(X,E) is studied. A function f in C(X,E) belongs to C*(X,E) if for any O-neighborhood U in E, there exists 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 [formula omitted] of X such that every function f in C (X,E) has an extension f in [formula omitted] when E is the space of all nationals, real numbers, complex numbers, or the real quaternions, [formula omitted] is just the space of all E-homomorphisms from C*(X,E) into E. This is also valid for a topological ring E which satisfies certain conditions. Also, two E*-compact spaces [Definition 10.1] X and Y are homeomorphic if, and only if, the rings C*(X,E) and C*(Y,E) are E-isomorphic, where E is any H*-topological ring [Definition 12.8].

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