The Relative Uniform Density of the Continuous Functions in the Baire Functions, and of a Divisible Archimedean -Group in any Epicompletion

For a subset A of an -group B, r(A,B) denotes the relative uniform closure of A in B. RX denotes the -group of all real-valued functions on the set X, and when X is a topological space, C∗(X) is the -group of all bounded continuous real-valued functions, and B(X) is the -group of all Baire functions. We show that B(X) = r (C∗(X), B(X)) = r ¡ C∗(X), R ¢ . This would appear to be a purely order-theoretic construction of B(X) from C(X) within RX . That result is then applied to the category Arch of archimedean -groups, and its subcategory W of -groups with distinguished weak unit. In earlier work we have described the epimorphisms of these categories, characterized those objects with no epic extension (called epicomplete), and for W, constructed all epic embeddings into epicomplete objects (epicompletions) using Baire functions. Now this apparatus is combined with the equation displayed above to make this contribution to the description of epimorphisms. In Arch orW, if a divisible -group A is epically embedded in an epicomplete -group B then B = r(A,B). Examples are presented to show that, in each of Arch andW, the hypothesis that B be epicomplete cannot be dropped. The paper is organized as follows. Section 1 contains basic definitions and facts, the statement of the main technical Theorem 1, and the derivation therefrom of the equation displayed in the abstract. Section 2 is devoted to the proof of Theorem 1, which involves some intricacies of the Baire classification of functions. Section 3 is a brief recollection of some of our results on epimorphisms and epicompletions in archimedean -groups. Section 4 is devoted to the issue of relative uniform density versus epimorphic embedding in the categoriesW and Arch respectively. The analysis forW is needed for Arch. ∗Department of Mathematics and Computer Science, University of Denver, Denver, Colorado 80208, U.S.A., email [email protected] †Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459, U.S.A., email [email protected]