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On Stone's theorem and the Axiom of Choice

Authors: C. Good, I. J. Tree and W. S. Watson
Journal: Proc. Amer. Math. Soc. 126 (1998), 1211-1218
MSC (1991): Primary 54D20; Secondary 03E25
MathSciNet review: 1425122
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Abstract: 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.

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Additional Information

C. Good

I. J. Tree
Affiliation: 62 Arle Gardens, Cheltenham, Gloucestershire GL51 8HR, England

W. S. Watson
Affiliation: Department of Mathematics, York University, North York, Ontario, Canada M3J 1P3

Received by editor(s): March 27, 1996
Received by editor(s) in revised form: September 17, 1996
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1998 American Mathematical Society