Vietoris topology

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Let $(X,\tau)$ be a topological space. The Vietoris topology for the hyperspace $\mathrm{CL}(X)$ is the smallest topology $\tau_v$ for $\mathrm{CL}(X)$ having the following property: $\{A \in \mathrm{CL}(X) \colon A \subset U\} \in \tau_v$ whenever $U \in \tau$, and $\{A \in \mathrm{CL}(X) \colon A \subset B \}$ is closed whenever $B$ is closed in $(X,\tau)$.


Illanes, Alejandro ; Nadler, Sam B., Jr. Hyperspaces. Fundamentals and recent advances. Monographs and Textbooks in Pure and Applied Mathematics, 216. Marcel Dekker, Inc., New York, 1999. xx+512 pp. ISBN: 0-8247-1982-4