The relative uniform density of the continuous functions in the Baire functions, and of a divisible Archimedean ℓ-group in any epicompletion

Abstract

For a subset A of an ℓ-group B, r( A, B) denotes the relative uniform closure of A in B. R X denotes the ℓ-group of all real-valued functions on the set X, and when X is a topological space, C ∗(X) is the ℓ-group of all bounded continuous real-valued functions, and B( X) is the ℓ-group of all Baire functions. We show that B(X)=r(C ∗(X),B(X))=r(C ∗(X),R X) . This would appear to be a purely order-theoretic construction of B( X) from C( X) within R X . That result is then applied to the category Arch of Archimedean ℓ-groups, and its subcategory W of ℓ-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 above to make this contribution to the description of epimorphisms. In Arch or W , if a divisible ℓ-group A is epically embedded in an epicomplete ℓ-group B then B= r( A, B). Examples are presented to show that, in each of Arch and W , the hypothesis that B be epicomplete cannot be dropped.