Uniform convergence of distance functionals under remetrization and infima of hyperspace topologies

TL;DR

This paper explores how distance functionals converge in hyperspaces under different metrics, introduces a new covering property, and establishes conditions for the existence of minimal convergence elements related to Heine-Borel metrics.

Contribution

It introduces a new covering property linked to bornologies, analyzes hyperspace convergence under various metrics, and proves the existence of minimal convergence elements for spaces with Heine-Borel metrics.

Findings

Introduces a new covering property for bornologies.

Establishes the existence of a minimal hyperspace convergence element.

Shows the minimal element exists iff the space has a compatible Heine-Borel metric.

Abstract

The objective of this paper is twofold. In the first half of the paper, we investigate upper parts of the hyperspace convergences determined by uniform convergence of distance functionals on a bornology under different metrizations of a metrizable space. To do this, a new covering property associated with the underlying bornology is introduced. An independent study of this new covering notion in relation to some well-known notions, such as strong uniform continuity, is also presented. In the second half, we study the infima of hyperspace convergences (induced by distance functionals) determined by a family of (uniformly) equivalent metrics. In particular, we establish the existence of the minimum element for the collection of upper Attouch-Wets convergences corresponding to all equivalent metrics on a metrizable space $X$. We show that such a minimum element exists if and only if $X$ has a compatible Heine-Borel metric. Our findings give several new insights into the theory of hyperspace convergences.