Vietoris topology

Summary

The Vietoris topology is a topology on a set of subsets of a topological space, usually the closed subsets or the compact subsets. A set is near a set $A$ when it lies inside a given open neighbourhood of $A$. It must also meet some given open sets that $A$ meets. It turns a family of subsets into a space, called a hyperspace. On the nonempty compact subsets of a metric space, it is the topology of the Hausdorff metric.