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Hypertopology

From Wikipedia, the free encyclopedia

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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

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 metric[3] and Vietoris topology.[4]

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Notation

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

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Examples

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Vietoris topology

Let be a closed subset and be a finite collection of open subsets of X. Define

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

Fell topology

A variant on the Vietoris topology is to allow only the sets where C is a compact subset of X and 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 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].

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 coincides the Fell topology.

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See also

References

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