The Samuel realcompactification

Abstract

For a uniform space (X, μ), we introduce a realcompactification of X by means of the family Uμ(X) of all the real-valued uniformly continuous functions on X, in the same way that the known Samuel compactification of the space is given by U∗μ(X) the set of all the bounded functions in Uμ(X). We will call it “the Samuel realcompactification” by several resemblances to the Samuel compactification. In this paper, we present different ways to construct such realcompactification as well as we study the corresponding problem of knowing when a uniform space is Samuel realcompact, that is, when it (topologically) coincides with its Samuel realcompactification. At this respect, we obtain as main result a theorem of Katětov–Shirota type, given in terms of a property of completeness, recently introduced by the authors, called Bourbaki-completeness.