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Admissible subrings of real-valued continuous functions Choo, Eng-Ung

Abstract

The object of this thesis is to study the relations between the algebraic properties of admissible subrings [Definition 0.4] of the ring C(X) of all real-valued continuous functions on X and the topological properties of the space X. Given an admissible subring G of C(X), there exists a unique G*-compactification [Definition 1.7 and 0.17] Y of X, with the properties that X is G*-embedded [Definition 0.16] in Y and G*(Y) [Definition 0.12] is admissible. If the cardinality |Y - X| of Y - X is finite or dG [Definition 2.1] is finite, then |Y — X| = dG. From Y, we can get the unique G-realcompactification [Definition 1.8] Z of X where X is G-embedded in Z, G(Z) is admissible and Z is G(Z)-realcompact [Definition 1.1]. It is proved that a G-realcompact space X and an H-realcompact space Y are homeomorphic if G and H are isomorphic admissible subrings. Let D(X) be the subring of all closed bounded functions in C(X). Then X is countably compact iff D(X) is admissible and closed under uniform convergence. For any admissible subring G of C(X), if dG is finite, then dG ≤ dD(X). Let B(X) be the subring of all bounded functions in C(X) which niap zero sets to closed sets in R. Then X is pseudocompact iff B(X) is admissible and closed under uniform convergence.

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