In the
mathematical branch of
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a hyperspace (or a space equipped with a hypertopology) is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, 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
In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A ch ...
is a
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
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.
See also
*
Hausdorff distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets o ...
*
Kuratowski convergence In mathematics, Kuratowski convergence or Painlevé-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlevé in lectures on mathematical analysis in 1902,This is reported in the Commen ...
*
Wijsman convergence
References
External links
Comparison of HypertopologiesHyperspacewiki
Topology
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