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

摘要

For a regularly locally compact topological space X of T_0 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 (X) of X. In this paper, we investigate properties of (X) and C((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.