For a regularly locally compact topological space $X$ of $\rm T_0$ separation axiom but not necessarily Hausdorff, we consider a map $\sigma$ 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 $\sigma$ is a topological embedding, and by taking the closure of $\sigma(X)$ 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.
Katsuhiko MATSUZAKI. "The Chabauty and the Thurston topologies on the hyperspace of closed subsets." J. Math. Soc. Japan 69 (1) 263 - 292, January, 2017. https://doi.org/10.2969/jmsj/06910263