Coincidence of Vietoris and Wijsman Topologies: A New Proof

Holá, L’.. "Coincidence of Vietoris and Wijsman Topologies: A New Proof." Serdica Mathematical Journal 23.3-4 (1997): 363-366. <http://eudml.org/doc/11623>.

@article{Holá1997,
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.},
author = {Holá, L’.},
journal = {Serdica Mathematical Journal},
keywords = {Vietoris Topology; Wijsman Topology; Metric Space; Compact Space; Vietoris topology; Wijsman topology},
language = {eng},
number = {3-4},
pages = {363-366},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Coincidence of Vietoris and Wijsman Topologies: A New Proof},
url = {http://eudml.org/doc/11623},
volume = {23},
year = {1997},
}

TY - JOUR
AU - Holá, L’.
TI - Coincidence of Vietoris and Wijsman Topologies: A New Proof
JO - Serdica Mathematical Journal
PY - 1997
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 23
IS - 3-4
SP - 363
EP - 366
AB - 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.
LA - eng
KW - Vietoris Topology; Wijsman Topology; Metric Space; Compact Space; Vietoris topology; Wijsman topology
UR - http://eudml.org/doc/11623
ER -