Vol. 68, No. 2, 1977

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Topologies on the set of closed subsets

Frank Arvey Wattenberg

Vol. 68 (1977), No. 2, 537–551

In this paper the techniques of Nonstandard Analysis are used to study topologies on the set Γ(X) of closed subsets of a topological space X. The first section of the paper investigates the “compact” topology developed by Narens and constructs a variant of that topology which is particularly useful for non locally compact spaces X. (When X is locally compact this variant is shown to be identical with Naren’s original “compact” topology.) This new topology is a natural extension to Γ(X) of the one point compactification of X embedded in Γ(X) in the obvious way with the point at infinity corresponding to the empty set. The second section shows that the techniques developed by Narens can be used to obtaln a naturaI characterization of the Vietoris Topology by considering monads of non nearstandard points. The final section uses this same approach to construct a topological analog of the Hausdorff metric for normal spaces.

Mathematical Subject Classification 2000
Primary: 54J05
Secondary: 54B20, 02H25
Received: 25 May 1976
Published: 1 February 1977
Frank Arvey Wattenberg