Geodesic
Coincidence of Vietoris and Wijsman Topologies: A New Proof
Serdica Mathematical Journal, Tome 23 (1997) no. 3-4, pp. 363-366.

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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.
Keywords: Vietoris Topology, Wijsman Topology, Metric Space, Compact Space 
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     title = {Coincidence of {Vietoris} and {Wijsman} {Topologies:} {A} {New} {Proof}},
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Holá, L’. Coincidence of Vietoris and Wijsman Topologies: A New Proof. Serdica Mathematical Journal, Tome 23 (1997) no. 3-4, pp. 363-366. http://geodesic.mathdoc.fr/item/SMJ2_1997_23_3-4_a11/