Topologies on Spaces of Sets

Abstract

Basic terminology from topology is reviewed, after which we introduce the Vietoris topology and two other topologies on the space of compact subsets of a topological space, as well as three parallel definitions of topologies on the space of closed subsets of the given space. Zorn’s lemma is used to prove a characterization of compactness due to Alexander, which is use to show that the space of compact subsets is Vietoris compact if the given space is compact. If the given space is metric, the Vietoris topology is induced by the Hausdorff distance between compact sets. Continuity properties of basic set operations, such as union, intersection, and cartesian product, are studied for these topologies, and we establish the continuity of the mapping of compact sets induced by a continuous function. It is shown that the union of the elements of a compact set of compacta is compact, and it is closed if the underlying space is regular.