Bourbaki-complete spaces and Samuel realcompactifiation

TL;DR

This thesis explores Bourbaki-completeness and Samuel realcompactification in uniform spaces, providing characterizations, embeddings, and conditions for metrizability and equivalence of compactifications.

Contribution

It offers new characterizations of Bourbaki-complete spaces, embeds them into universal spaces, and characterizes Samuel realcompact spaces in terms of uniform partitions and measurability.

Findings

Bourbaki-completeness is equivalent to completeness for star-finite base uniformities.

Characterization of Samuel realcompact spaces as Bourbaki-complete with no Ulam-measurable partitions.

Identification of conditions under which Samuel and Hewitt realcompactifications coincide.

Abstract

This thesis belongs to the area of General Topology and, in particular, to the field of study of uniform spaces. It is divided in three parts where the related topics Bourbaki-completeness and Samuel realcompactification are studied. In the first part of the thesis we study Bourbaki-completeness and cofinal Bourbaki-completeness (equivalently, uniform strong-paracompactness) of uniform spaces. In particular, we solve several primary problems related to products, subspaces, hyperspaces, metric spaces and fine spaces. Moreover, we show that completeness and cofinal completeness are, respectively, equivalent to Bourbaki-completeness and cofinal Bourbaki-completeness whenever we consider uniformities having a base of star-finite covers. In the second part, we embed the Bourbaki-complete metric spaces (and after, the Bourbaki-complete uniform spaces) into an universal space. Next, we characterized those metric spaces which are metrizable by a Bourbaki-complete metric or cofinally Bourbaki-complete metric. In the third part we characterize the Samuel realcompact spaces, that is, those uniform spaces which are complete with the weak uniformity induced by all the real-valued uniformly continuous functions on the space. It turns out that these are exactly the Bourbaki-complete uniform spaces having no uniform partition of Ulam-measurable cardinal. Finally, as a natural step in order to generalize the above result, we study which kind of metric or uniform spaces satisfy that the Samuel realcompactification and the Hewitt realcompactification are equivalent.