# The Samuel realcompactification of a metric space

## TL;DR

This paper introduces the Samuel realcompactification for metric spaces, characterizes Samuel realcompactness via Bourbaki-completeness and non-measurable cardinality of discrete subspaces, and explores its relation to other compactifications.

## Contribution

It defines the Samuel realcompactification for metric spaces and characterizes Samuel realcompactness using Bourbaki-completeness and cardinality conditions, extending classical results.

## Key findings

- A metric space is Samuel realcompact iff it is Bourbaki-complete and all uniformly discrete subspaces have non-measurable cardinal.
- A finite-dimensional normed space is Samuel realcompact, but realcompactness and Samuel realcompactness can differ significantly.
- The paper relates the Samuel realcompactification to Lipschitz, Hewitt-Nachbin, and completion compactifications.

## Abstract

In this paper we introduce a realcompactification for any metric space (X, d), defined by means of the family of all its real-valued uniformly continuous functions. We call it the Samuel realcompactification, according to the well known Samuel compactification associated to the family of all the bounded real-valued uniformly continuous functions. Among many other things, we study the corresponding problem of the Samuel realcompactness for metric spaces. At this respect, we prove that a result of Kat\v{e}tov-Shirota type occurs in this context, where the completeness property is replaced by Bourbaki-completeness (a notion recently introduced by the authors) and the closed discrete subspaces are replaced by the uniformly discrete ones. More precisely, we see that a metric space (X, d) is Samuel realcompact iff it is Bourbaki-complete and every uniformly discrete subspace of X has non-measurable cardinal. As a consequence, we derive that a normed space is Samuel realcompact iff it has finite dimension. And this means in particular that realcompactness and Samuel realcompactness can be very far apart. The paper also contains results relating this realcompactification with the so-called Lipschitz realcompactification (also studied here), with the classical Hewitt-Nachbin realcompactification and with the completion of the initial metric space.