The Baire category theorem and choice (Q1588332)

The purpose of this note is to investigate what weakening of the Axiom of Choice will suffice, or is needed, to prove various forms of the Baire Category theorem. The first section of the paper includes a survey of known relationships between Axiom of Choice type statements and Baire Category theorems for various classes of spaces. The authors highlight the following examples: no Choice is needed to prove that compact pseudometric spaces are Baire spaces; the Axiom of Countable Choice is equivalent to the statement that countable products of compact pseudometric spaces are Baire spaces; the Axiom of Dependent Choice is equivalent to the statement that countable products of compact Hausdorff spaces are Baire spaces. A space is Baire if countable intersections of dense open sets are not empty. The Axiom of Countable Choice is the statement that countable products of non-empty sets are not empty. The Axiom of Dependent Choice is the statement that for each set \(X\) and binary relation \(R\subset X\times X\) such that \(R\cap (\{x\}\times X)\neq \emptyset\) for each \(x\in X\), there is a sequence \(\{x_n : n\in \omega\}\subset X\) such that \((x_n, x_{n+1})\in R\) for all \(n\).