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

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.