Proximally well-monotone covers and $QH$-singularity

TL;DR

This paper explores the use of well-monotone covers in quasi-uniform spaces to identify conditions under which different quasi-uniformities induce identical topologies on the hyperspace, contributing to the understanding of $QH$-singularity.

Contribution

It introduces a new approach using proximally well-monotone covers to analyze $QH$-singularity in quasi-uniform spaces, providing novel criteria for when distinct quasi-uniformities share the same hyperspace topology.

Findings

Identifies conditions for $QH$-singularity using well-monotone covers.

Establishes relationships between quasi-uniformities and their hyperspace topologies.

Provides examples illustrating the theory.

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.