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Let (X, d) be a metric space and CL(X) the family of all
nonempty closed subsets of X. We provide a new proof of the fact that the
coincidence of the Vietoris and Wijsman topologies induced by the metric
d forces X to be a compact space. In the literature only a more involved
and indirect proof using the proximal topology is known. Here we do not
need this intermediate step. Moreover we prove that (X, d) is boundedly
compact if and only if the bounded Vietoris and Wijsman topologies on
CL(X) coincide....
Let X be a Tikhonov space, C(X) be the space of all continuous real-valued functions defined on X, and CL(X×ℝ) be the hyperspace of all nonempty closed subsets of X×ℝ. We prove the following result: Let X be a locally connected locally compact paracompact space, and let F ∈ CL(X×ℝ). Then F is in the closure of C(X) in CL(X×ℝ) with the Vietoris topology if and only if: (1) for every x ∈ X, F(x) is nonempty; (2) for every x ∈ X, F(x) is connected; (3) for every isolated x ∈ X, F(x) is a singleton...
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