Geodesic
Normal Vietoris implies compactness: a short proof
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 181-182.

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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.
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},
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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/