Hypertopology

In the mathematical branch of topology, a hyperspace (or a space equipped with a hypertopology) is a topological space, which consists of the set CL(X) of all non-empty closed subsets of another topological space X, equipped with a topology so that the canonical map

$$i : x \mapsto \overline{\{x\}},$$

is a homeomorphism onto its image. As a consequence, a copy of the original space X lives inside its hyperspace CL(X).

Early examples of hypertopology include the Hausdorff metric and Vietoris topology.