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.

Notation

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

Examples

Vietoris topology

Let $F$ be a closed subset and $U_1,\dots ,U_n$ be a finite collection of open subsets of X. Define

$V(F,U_1,\dots ,U_n)=\{Y\subseteq X\text{ closed}\mid Y\cap F=\mathrm{\varnothing }\text{ and }Y\cap U_i\ne \mathrm{\varnothing }\text{ for every }1\le i\le n\}.$

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

Fell topology

A variant on the Vietoris topology is to allow only the sets $V(C,U_1,\dots ,U_n)$ where C is a compact subset of X and $U_1,\dots ,U_n$ 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. Note, though, that the canonical map $i:x\mapsto \overline{\{x\}}$ is a homeomorphism onto its image if and only if X is Hausdorff, 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. 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.

Other constructions

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.

See also

• Hausdorff distance
• Kuratowski convergence
• Wijsman convergence
• Chabauty topology

External links

• Comparison of Hypertopologies
• Hyperspacewiki