The Baire category theorem in products of linear spaces and topological groups
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)].
completely metrizable linear topological space
path connected abelian topological group
scientific article; zbMATH DE number 3947183
scientific article