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 metric[3] and Vietoris topology.[4]

See also

• Hausdorff distance
• Kuratowski convergence
• Wijsman convergence

References

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.

External links

• Comparison of Hypertopologies
• Hyperspacewiki