Remark 1

Let G be a topological group and let A be a meagre subset of G. Suppose that A contains a nonempty open subset U of G and let $y$ be a point of U. Then U is meagre and for every $x\in G,xy^{-1}U$ is a neighbourhood of $x$ in G. Hence every point of G possesses a meagre neighbourhood, therefore G is meagre (IV, p. 360, lemma 1).

Conversely, if G is not meagre, every meagre subset has empty interior and G is a Baire space. Since a Baire space is not meagre, this proves the following assertion: In order that a topological group be a Baire space, it is necessary and sufficient that it be not meagre.