Uniformizable and realcompact bornological universes

TL;DR

This paper characterizes when bornological universes are uniformizable and introduces the concept of realcompactness for them, extending previous results and providing new characterizations.

Contribution

It extends Hu's metrizability result to uniformizability and introduces realcompactness for bornological universes with multiple characterizations.

Findings

Characterization of uniformizable bornological universes

Definition of realcompactness for bornological universes

Various characterizations of the new concept

Abstract

Bornological universes were introduced some time ago by Hu and obtained renewed interest in recent articles on convergence in hyperspaces and function spaces and optimization theory. One of Hu's results gives us a necessary and sufficient condition for which a bornological universe is metrizable. In this article we will extend this result and give a characterization of uniformizable bornological universes. Furthermore, a construction on bornological universes that the author used to find the bornological dual of function spaces endowed with the bounded-open topology will be used to define realcompactness for bornological universes. We will also give various characterizations of this new concept.