Vietoris topology

Remarks

The same subbasis, intersected with a smaller family, defines the Vietoris topology on that family, for example on the nonempty compact subsets $\mathrm{K}(X)$. If $X$ is a metric space, the Vietoris topology on $\mathrm{K}(X)$ is the topology of the Hausdorff metric. If $X$ is compact Hausdorff, then $\mathrm{CL}(X)$ with the Vietoris topology is compact Hausdorff. If $X$ is $T_1$, the map $x \mapsto \set{x}$ is a topological embedding of $X$ into $\mathrm{CL}(X)$.

Example: let $X = [0, 1]$, let $A = [0, 1]$, and let $A_n = \set{k/n : 0 \le k \le n}$. Each basic neighbourhood of $A$ has the form $U^+ \cap V_1^- \cap \cdots \cap V_m^-$ with $A \subseteq U$ and each $V_i$ open and nonempty. For large $n$, the set $A_n$ meets each $V_i$ and lies in $U = X$, so $A_n \to A$.

Non-example: the sets $\set{0}$ do not converge to $[0, 1]$, though they lie in every $U^+$ with $[0, 1] \subseteq U$. The open set $V = (1/2, 1]$ gives a neighbourhood $V^-$ of $[0, 1]$ that does not contain $\set{0}$.

The resulting space is called a hyperspace of $X$, and the topology is also called the finite topology or exponential topology.