The Baire Category Theorem and choice

Abstract

The status of the Baire Category Theorem in ZF (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice) is investigated. Typical results: 1. The Baire Category Theorem holds for compact pseudometric spaces. 2. The Axiom of Countable Choice is equivalent to the Baire Category Theorem for countable products of compact pseudometric spaces. 3. The Axiom of Dependent Choice is equivalent to the Baire Category Theorem for countable products of compact Hausdorff spaces. 4. The Baire Category Theorem for B -compact regular spaces is equivalent to the conjunction of the Axiom of Dependent Choice and the Weak Ultrafilter Theorem .