arXiv subjects
Publications and source records attributed to Tom Vroegrijk.
In this paper we use a certain class of well-monotone covers on a quasi-uniform space $(X, \mathcal{U})$ to investigate whether there are quasi-uniformities $\mathcal{V}$ that are distinct from $\mathcal{U}$, but have the property that the associated Hausdorff quasi-uniformities $\mathcal{U}_H$ and $\mathcal{V}_H$ on the hyperspace of $X$ have the same underlying topologies.
The bornological convergence structures that have been studied recently as generalizations of Attouch-Wets convergence define pretopologies on hyperspaces. In this paper we characterize the topological reflections of these pretopologies and translate the constructions necessary to define bornological convergence to a broader spectrum of hyperspace topologies.
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.