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

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

is a homeomorphism onto its image. As a consequence, a copy of the original space X lives inside its hyperspace CL(X).^{1 2}

Early examples of hypertopology include the Hausdorff metric³ and Vietoris topology.⁴

Various notation is used by different authors to denote the set of all closed subsets of a topological space X, including CL(X), ${\displaystyle {\mathcal {F}}(X)}$ and ${\displaystyle 2^{X}}$.

Let ${\displaystyle F}$ be a closed subset and ${\displaystyle U_{1},\ldots ,U_{n}}$ be a finite collection of open subsets of X. Define

${\displaystyle V(F,U_{1},\ldots ,U_{n})=\{Y\subseteq X{\text{ closed}}\mid Y\cap F=\emptyset {\text{ and }}Y\cap U_{i}\neq \emptyset {\text{ for every }}1\leq i\leq n\}.}$

These sets form a basis for a topology on CL(X), called the Vietoris or finite topology, named for Leopold Vietoris.⁵

A variant on the Vietoris topology is to allow only the sets ${\displaystyle V(C,U_{1},\ldots ,U_{n})}$ where C is a compact subset of X and ${\displaystyle U_{1},\ldots ,U_{n}}$ a finite collection of open subsets. This is again a base for a topology on CL(X) called the Fell topology or the H-topology.⁶ Note, though, that the canonical map ${\displaystyle i:x\mapsto {\overline {\{x\}}}}$ is a homeomorphism onto its image if and only if X is Hausdorff,⁷ so for non-Hausdorff X, the Fell topology is not a hypertopology in the sense of this article.

The Vietoris and Fell topologies coincide if X is a compact space, but have quite different properties if not. For instance, the Fell topology is always compact and it is compact Hausdorff whenever if X is locally compact.⁸ On the other hand, the Vietoris topology is compact if and only if X is compact and Hausdorff if and only if X is regular.⁹

The Hausdorff distance on the closed subsets of a bounded metric space X induces a topology on CL(X). If X is a compact metric space, this agrees with the Vietoris and Fell topologies.

The Chabauty topology on the closed subsets of a locally compact group coincides with the Fell topology.