Title: 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 Katětov-Shirota type, by means of a new property of completeness
recently introduced by the authors, called Bourbaki-completeness.