Topologies of closed subsets

Abstract:

In this paper various topologies on closed subsets of a topological space are considered. The interrelationships between these topologies are explored, and several applications are given. The methods of proof as well as some intrinsic definitions assume a familiarity with A. Robinson’s nonstandard analysis. E. Michael (Topologies of spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182), K. Kuratowski (Topology, Vols. I and II, Academic Press, New York, 1968), L. Vietoris (Berichezweiter Ordnung, Monatsh. Math.-Phys. 33 (1923), 49-62), and others have considered methods of putting topologies on closed subsets of a topological space. These topologies have the property that if the underlying topological space is compact then the topology of closed subsets is also compact. In general, however, these topologies of closed subsets are not compact. In this paper, a topology of closed subsets of a topological space is constructed that is always compact. This topology is called the compact topology and has many pleasant features. For closed subsets of compact Hausdorff spaces, this topology agrees with Vietoris’ topology. For arbitrary spaces, there are interesting connections between the compact topology and topological convergence of subsets, including generalized versions of the Bolzano-Weierstrass theorem.