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Uncountable Discrete Sets in Extensions and Metrizability

Published online by Cambridge University Press:  20 November 2018

Murray Bell and
John Ginsburg
Murray Bell
Affiliation:
University of Manitoba, Winnipeg Manitoba, Canada R3T 2N2
John Ginsburg
Affiliation:
University of Winnipeg, Winnipeg Manitoba, Canada R3B 2E9
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Abstract

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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.

Keywords

54A2554E3554B1054B2054D3002K05uncountable discrete setspace of closed setssuperextensionsquaremetrizabilitycompactnessMartin's axiom

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 4 , 01 December 1982, pp. 472 - 477
Copyright
Copyright © Canadian Mathematical Society 1982

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