Set convergences and uniform convergence of distance functionals on a bornology

TL;DR

This paper studies the topological properties of hyperspaces of closed sets in metric spaces using a unified approach to set convergence via distance functionals, introducing new bornologies and metrics.

Contribution

It introduces new classes of bornologies and a new metric topology on hyperspaces, extending classical convergence topologies like Wijsman and Hausdorff.

Findings

Characterization of topological properties of hyperspaces

Introduction of new bornologies and metrics

Extension of classical convergence topologies

Abstract

For a metric space $(X,d)$, Beer, Naimpally, and Rodriguez-Lopez in ([17]) proposed a unified approach to explore set convergences via uniform convergence of distance functionals on members of an arbitrary family $\mathcal{S}$ of subsets of $X$. The associated topology on the collection $CL(X)$ of all nonempty closed subsets of $(X,d)$ is denoted by $\tau_{\mathcal{S},d}$. As special cases, this unified approach includes classical Wijsman, Attouch-Wets, and Hausdorff distance topologies. In this article, we investigate various topological characteristics of the hyperspace $(CL(X), \tau_{\mathcal{S},d})$ when $\mathcal{S}$ is a bornology on $(X,d)$. In order to do this, a new class of bornologies and a new metric topology on $CL(X)$ have been introduced and studied.