The Chabauty and the Thurston topologies on the hyperspace of closed subsets

Abstract

For a regularly locally compact topological space X of T₀ separation axiom but not necessarily Hausdorff, we consider a map σ from X to the hyperspace C(X) of all closed subsets of X by taking the closure of each point of X. By providing the Thurston topology for C(X), we see that σ is a topological embedding, and by taking the closure of σ(X) with respect to the Chabauty topology, we have the Hausdorff compactification $\widehat X$ of X. In this paper, we investigate properties of $\widehat X$ and C($\widehat X$) equipped with different topologies. In particular, we consider a condition under which a self-homeomorphism of a closed subspace of C(X) with respect to the Chabauty topology is a self-homeomorphism in the Thurston topology.