The Baire category theorem and choice
type: Item
instance of: scholarly article
title: The Baire category theorem and choice
zbMATH Open document ID: 0991.54036
DOI: 10.1016/S0166-8641(99)00132-7
author: Horst Herrlich
author: Kyriakos Keremedis
published in: Topology and its Applications
publication date: 2002-08-28T00:00:00Z
review text: 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 \(n\).
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 03E25
zbMATH DE Number: 1539279
zbMATH Keywords: axiom of dependent choice
zbMATH Keywords: Baire category theorem
zbMATH Keywords: countable Tikhonov theorem
reviewed by: Alan Dow
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1539279