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The Baire category theorem and choice. (English) Zbl 0991.54036

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 \(n\).

MSC:

54E52 Baire category, Baire spaces
54A35 Consistency and independence results in general topology
54D30 Compactness
03E25 Axiom of choice and related propositions
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