On Stone's theorem and the axiom of choice

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  • School of Mathematics and Statistics
  • York University


It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's Theorem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forcing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice.


Original languageEnglish
Pages (from-to)1211-1218
Number of pages8
JournalProceedings of the American Mathematical Society
Issue number4
Publication statusPublished - 1998

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