Proximally well-monotone covers and $QH$-singularity

Abstract

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.