The Baire category theorem in products of linear spaces and topological groups
type: Item
instance of: scholarly article
title: The Baire category theorem in products of linear spaces and topological groups
zbMATH Open document ID: 0589.54040
DOI: 10.1016/0166-8641(86)90025-8
author: Jan van Mill
author: R. Pol
published in: Topology and its Applications
publication date: 1986-01-01T00:00:00Z
review text: It has been difficult to construct examples of two Baire spaces whose products are not Baire spaces. This paper gives additional constructions of such examples which furthermore have linear or group structures. In particular, if X is a nonseparable completely metrizable linear topological space, then X contains linear subspaces E and F which are Baire spaces while \(E\times F\) is not a Baire space. In addition, if X has weight \(\chi_ 1\), then E and F may be chosen so that X is the direct sum of E and F. In a set of parallel but different constructions, it is also established that every path connected abelian topological group contains two subgroups which are Baire spaces while their product is not a Baire space. In the proof, ''independent'' Cantor sets are constructed as by [\textit{J. Mycielski} in Fundam. Math. 55, 139-147 (1964; Zbl 0124.012)]. Then the general method of construction is along the lines of the Fleissner-Kunen construction [\textit{W. G. Fleissner} and \textit{K. Kunen}, Fundam. Math. 101, 229-240 (1978; Zbl 0413.54036)].
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 3947183
zbMATH Keywords: Baire spaces
zbMATH Keywords: completely metrizable linear topological space
zbMATH Keywords: path connected abelian topological group
zbMATH Keywords: Cantor sets
reviewed by: R. A. McCoy
MaRDI profile type: Publication
full work available at URL: 0166-8641(86)90025-8
OpenAlex ID: W2015603773
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description: scientific article; zbMATH DE number 3947183