# On Stone's theorem and the axiom of choice

Research output: Contribution to journal › Article

## Authors

## Colleges, School and Institutes

## External organisations

- School of Mathematics and Statistics
- York University

## 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.

## Details

Original language | English |
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Pages (from-to) | 1211-1218 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 126 |

Issue number | 4 |

Publication status | Published - 1998 |