Geodesic
Normal Vietoris implies compactness: a short proof
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 181-182 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 54B20, 54D30, 54E15
Keywords: hyperspaces; Vietoris topology; locally finite topology; Hausdorff metric; compactness; normality; countable compactness 
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     title = {Normal {Vietoris} implies compactness: a short proof},
     journal = {Czechoslovak Mathematical Journal},
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Maio, G. Di; Meccariello, E.; Naimpally, S. Normal Vietoris implies compactness: a short proof. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 181-182. http://geodesic.mathdoc.fr/item/CMJ_2004_54_1_a14/