On normality of the Wijsman topology

TL;DR

This paper investigates the conditions under which the Wijsman topology on the hyperspace of a metric space is normal, establishing that for linear metric spaces, normality is equivalent to metrizability.

Contribution

It provides a partial answer to whether normality of the Wijsman topology implies its metrizability, specifically proving the equivalence in linear metric spaces.

Findings

Normality of Wijsman topology on linear metric spaces is equivalent to metrizability.

Partial answer to the normality-metrizability question for general metric spaces.

Additional results on the normality conditions of the Wijsman topology.

Abstract

Let $(X,\rho)$ be a metric space and $(CL(X),W_\rho)$ be the hyperspace of all nonempty closed subsets of $X$ equipped with the Wijsman topology. The Wijsman topology is one of the most important classical hyperspace topologies. We give a partial answer to a question posed in \cite{maio} whether the normality of $(CL(X),W_\rho)$ is equivalent to its metrizability. If $(X,\rho)$ is a linear metric space, then $(CL(X),W_\rho)$ is normal if and only if $(CL(X),W_\rho)$ is metrizable. Some further results concerning normality of the Wijsman topology on $CL(X)$ are also proved.