# Vietoris topology

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 $(X,\tau)$.