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).^[1] ^[2]

Early examples of hypertopology include the Hausdorff metric^[3] and Vietoris topology.^[4]

Notation

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

Examples

Vietoris topology

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

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

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

Fell topology

A variant on the Vietoris topology is to allow only the sets $V(C,U_{1},\ldots ,U_{n})$ where C is a compact subset of X and $U_{1},\ldots ,U_{n}$ 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 $i:x\mapsto {\overline {\{x\}}}$ 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]

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.

References

^ Lucchetti, Roberto; Angela Pasquale (1994). "A New Approach to a Hyperspace Theory" (PDF). Journal of Convex Analysis. 1 (2): 173–193. Retrieved 20 January 2013.
^ Beer, G. (1994). Topologies on closed and closed convex sets. Kluwer Academic Publishers.
^ Hausdorff, F. (1927). Mengenlehre. Berlin and Leipzig: W. de Gruyter.
^ Vietoris, L. (1921). "Stetige Mengen". Monatshefte für Mathematik und Physik. 31: 173–204. doi:10.1007/BF01702717.
^ Vietoris, L. (1921). "Stetige Mengen". Monatshefte für Mathematik und Physik. 31: 173–204. doi:10.1007/BF01702717.
^ Fell, J. M. G. (1962). "A Hausdorff topology for the closed subsets of a locally compact non- Hausdorff space". Proc. Am. Math. Soc. 13: 472–476.
^ Fell, J. M. G. (1962). "A Hausdorff topology for the closed subsets of a locally compact non- Hausdorff space". Proc. Am. Math. Soc. 13: 472–476.
^ Fell, J. M. G. (1962). "A Hausdorff topology for the closed subsets of a locally compact non- Hausdorff space". Proc. Am. Math. Soc. 13: 472–476.
^ Michael, Ernest (1951). "Topologies on spaces of subsets". Trans. Am. Math. Soc. 71: 152–182.

External links

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[1] Lucchetti, Roberto; Angela Pasquale (1994). "A New Approach to a Hyperspace Theory" (PDF). Journal of Convex Analysis. 1 (2): 173–193. Retrieved 20 January 2013.

[2] Beer, G. (1994). Topologies on closed and closed convex sets. Kluwer Academic Publishers.

[3] Hausdorff, F. (1927). Mengenlehre. Berlin and Leipzig: W. de Gruyter.

[4] Vietoris, L. (1921). "Stetige Mengen". Monatshefte für Mathematik und Physik. 31: 173–204. doi:10.1007/BF01702717.

[5] Vietoris, L. (1921). "Stetige Mengen". Monatshefte für Mathematik und Physik. 31: 173–204. doi:10.1007/BF01702717.

[6] Fell, J. M. G. (1962). "A Hausdorff topology for the closed subsets of a locally compact non- Hausdorff space". Proc. Am. Math. Soc. 13: 472–476.

[7] Fell, J. M. G. (1962). "A Hausdorff topology for the closed subsets of a locally compact non- Hausdorff space". Proc. Am. Math. Soc. 13: 472–476.

[8] Fell, J. M. G. (1962). "A Hausdorff topology for the closed subsets of a locally compact non- Hausdorff space". Proc. Am. Math. Soc. 13: 472–476.

[9] Michael, Ernest (1951). "Topologies on spaces of subsets". Trans. Am. Math. Soc. 71: 152–182.

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