The Samuel realcompactification

Abstract

For a uniform space (X, $\mu$), we introduce a realcompactification of X by means of the family $U_{\mu}(X)$ of all the real-valued uniformly continuous functions, in the same way that the known Samuel compactification is given by $U^{*}_{\mu}(X)$ the set of all the bounded functions in $U_{\mu}(X)$. We will call it "the Samuel realcompactification" by several resemblances to the Samuel compactification. In this note, 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, it coincides with its Samuel realcompactification. At this respect we obtain as main result a theorem of Kat\v{e}tov-Shirota type, by means of a new property of completeness recently introduced by the authors, called Bourbaki-completeness.