Normal Vietoris implies compactness: a short proof

G. Di Maio; E. Meccariello; Somashekhar Naimpally

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 1, page 181-182
  • ISSN: 0011-4642

Abstract

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One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities.

How to cite

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Maio, G. Di, Meccariello, E., and Naimpally, Somashekhar. "Normal Vietoris implies compactness: a short proof." Czechoslovak Mathematical Journal 54.1 (2004): 181-182. <http://eudml.org/doc/30847>.

@article{Maio2004,
abstract = {One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities.},
author = {Maio, G. Di, Meccariello, E., Naimpally, Somashekhar},
journal = {Czechoslovak Mathematical Journal},
keywords = {hyperspaces; Vietoris topology; locally finite topology; Hausdorff metric; compactness; normality; countable compactness; hyperspaces; Vietoris topology; locally finite topology; Hausdorff metric; compactness; normality; countable compactness},
language = {eng},
number = {1},
pages = {181-182},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Normal Vietoris implies compactness: a short proof},
url = {http://eudml.org/doc/30847},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Maio, G. Di
AU - Meccariello, E.
AU - Naimpally, Somashekhar
TI - Normal Vietoris implies compactness: a short proof
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 181
EP - 182
AB - One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities.
LA - eng
KW - hyperspaces; Vietoris topology; locally finite topology; Hausdorff metric; compactness; normality; countable compactness; hyperspaces; Vietoris topology; locally finite topology; Hausdorff metric; compactness; normality; countable compactness
UR - http://eudml.org/doc/30847
ER -

References

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  8. On spaces of closed subsets, Sibirskii Matem.  Z. 16 (1975), 627–629. (1975) 

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