Uniformizable and realcompact bornological universes

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.

💡 Research Summary

The paper investigates two fundamental extensions of Hu’s theory of bornological universes: uniformizability and realcompactness. A bornological universe (X, B) consists of a topological space X together with a bornology B (a family of subsets closed under taking singletons, subsets, and finite unions). Hu’s earlier metrization theorem states that (X, B) is metrizable precisely when X is metrizable, B has a countable base, and every B₁∈B is contained in the interior of some B₂∈B.

The author first tackles the question “when is a bornological universe uniformizable?” Two natural bornologies arise from any uniform space (X, U): the Bourbaki‑bounded sets and the totally bounded sets. Accordingly, the paper defines:

• bb‑uniformizable: there exists a uniformity U on X such that B coincides with the family of Bourbaki‑bounded subsets of (X, U);
• tb‑uniformizable: there exists a uniformity U on X such that B coincides with the family of totally bounded subsets of (X, U).

Theorem 4 (citing