Wijsman hyperspaces: Subspaces and embeddings (Q408574)

Given a metric space \((X,d)\) the hyperspace \(2^X\) consisting of all nonempty closed subsets is equipped with the Wijsman Topology \(\mathcal{T}_{w(d)}\). If on \(2^X\) one considers the source of all distance functions \(d(.,S)\), for \(S\) running through all nonempty closed subsets of \(X\) then the Wijsman topology is the initial lift in \textbf{Top} of this source. In this paper the existence of isolated points in the hyperspace is studied. Moreover it is shown that every Tychonoff space can be embedded as a closed subspace in the Wijsman hyperspace of a complete metric space which is locally \(\mathbb{R}\).