The Baire category theorem in products of linear spaces and topological groups

Abstract

A space is a Baire space if the intersection of countably many dense open sets is dense. We show that if X is a non-separable completely metrizable linear space (pathconnected abelian topological group) then X contains two linear subspaces (subgroups) E and F such that both E and F are Baire but E×F is not. If X is a completely metrizable linear space of weight ℵ1 then X is the direct sum E⊕F of two linear subspaces E and F such that both E and F are Baire but E×F is not.