# Coincidence of Vietoris and Wijsman Topologies: A New Proof

Serdica Mathematical Journal (1997)

- Volume: 23, Issue: 3-4, page 363-366
- ISSN: 1310-6600

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topHolá, 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 -

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