Geodesic
Uncountable Discrete Sets in Extensions and Metrizability
Canadian mathematical bulletin, Tome 25 (1982) no. 4, pp. 472-477

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If X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.
DOI : 10.4153/CMB-1982-068-3
Mots-clés : 54A25, 54E35, 54B10, 54B20, 54D30, 02K05, uncountable discrete set, space of closed sets, superextension, square, metrizability, compactness, Martin's axiom 
Bell, Murray; Ginsburg, John. Uncountable Discrete Sets in Extensions and Metrizability. Canadian mathematical bulletin, Tome 25 (1982) no. 4, pp. 472-477. doi: 10.4153/CMB-1982-068-3
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     title = {Uncountable {Discrete} {Sets} in {Extensions} and {Metrizability}},
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