Vietoris Topology Wijsman Topology Metric Space Compact Space
Issue Date:
1997
Publisher:
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Mathematical Journal, Vol. 23, No 3-4, (1997), 363p-366p
Abstract:
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.
