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1 results

  • Uncountable Discrete Sets in Extensions and Metrizability

  • Murray Bell, John Ginsburg
  • Journal:
    Canadian Mathematical Bulletin / Volume 25 / Issue 4 / 01 December 1982
    Published online by Cambridge University Press:
    20 November 2018, pp. 472-477
    Print publication:
    01 December 1982
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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.