On normality of the Wijsman topology

Abstract

Let (X, ρ) be a metric space and (CL(X), W ρ ) 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 Di Maio (Quaderni di Matematica, 3:55–92, 1998) whether the normality of (CL(X), W ρ ) is equivalent to its metrizability. If (X, ρ) is a linear metric space, then (CL(X), W ρ ) is normal if and only if (CL(X), W ρ ) is metrizable. Some further results concerning normality of the Wijsman topology on CL(X) are also proved.