# The Samuel realcompactification

## TL;DR

This paper introduces the Samuel realcompactification for uniform spaces, exploring its construction, properties, and conditions under which a space coincides with its realcompactification, highlighting a new completeness property called Bourbaki-completeness.

## Contribution

It defines the Samuel realcompactification, provides multiple construction methods, and characterizes when a uniform space is Samuel realcompact using Bourbaki-completeness.

## Key findings

- Established the Samuel realcompactification as a new concept.
- Connected Samuel realcompactification with Bourbaki-completeness.
- Proved a Katětov-Shirota type theorem for uniform spaces.

## 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.