powerspace

Powerspace is a term used in topology to denote the hyperspace of a topological space X, typically the set of all closed subsets of X (often restricted to nonempty closed subsets). The powerspace is equipped with the Vietoris topology, also known as the hyperspace topology, which encodes convergence of closed sets and interacts with standard set-theoretic operations.

The Vietoris topology on the powerspace is generated by a subbasis consisting of two kinds of sets for each open subset U of X: the upper sets [U] = {F in CL(X) : F ⊆ U} and the lower sets ⟨U⟩ = {F in CL(X) : F ∩ U ≠ ∅}. Finite intersections of these subbasic sets form a basis for the topology. This structure makes the powerspace a natural setting for studying how collections of points in X behave as sets.

If X is compact, the hyperspace CL(X) with the Vietoris topology is compact; when focusing on nonempty compact subsets, the corresponding hyperspace K(X) is also compact. In metric spaces, the Vietoris topology agrees with the topology induced by the Hausdorff metric on nonempty closed sets, yielding a metrizable hyperspace when X is compact metric.

Variants exist: some authors use the set of all closed subsets, others restrict to nonempty, and some distinguish between closed and compact subsets. The term powerspace is often used interchangeably with hyperspace in this context.

Applications of the powerspace include analysis of convergence of families of sets, continuity properties of set-valued maps, and roles in continuum theory, domain theory, and constructive topology. The construction is functorial: a continuous map f: X → Y induces a continuous map between the corresponding powerspaces.