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Countably Compact Spaces and Martin's Axiom

Published online by Cambridge University Press:  20 November 2018

William Weiss
William Weiss*
Affiliation:
University of Toronto, Toronto, Ontario
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The relationship between compact and countably compact topological spaces has been studied by many topologists. In particular an important question is: “What conditions will make a countably compact space compact?” Conditions which are “covering axioms” have been extensively studied. The best results of this type appear in [19]. We wish to examine countably compact spaces which are separable or perfectly normal. Recall that a space is perfect if and only if every closed subset is a Gδ, and that a space is perfectly normal if and only if it is both perfect and normal. We show that the following statement follows from MA +┐ CH and thus is consistent with the usual axioms of set theory: Every countably compact perfectly normal space is compact. This result is Theorem 3 and can be understood without reading much of what goes before.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 2 , 01 April 1978 , pp. 243 - 249
Copyright
Copyright © Canadian Mathematical Society 1978

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