Vietoris topology

Definition

The Vietoris topology is a topology defined on the set of all non-empty subsets of a topological space based on the topology on the original topological space. The topology is defined by the following basis:

For every non-empty finite collection of non-empty open subsets $U_{1},U_{2},\dots ,U_{n}$ in $X$, the corresponding basis element is the set of all non-empty subsets $A$ of $X$ for which all the intersections $A\cap U_{i}$ are non-empty.