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.
