Lemma 4

Let X be a Baire space, let G be a topological group acting continuously on X and let B be an approachable subset of X which is not meagre. The set of points $g\in G$ such that $B\cap gB=\not\emptyset$ is a neighbourhood of the identity element of G.

Since B is approachable, $\complement B$ is also approachable, for the set of approachable subsets of X is a tribe. Hence there exists an open subset U of X such that $U\cap \complement B$ and $B\cap \complement U$ are meagre in X.

Let V be a neighbourhood of the identity element of G and W a nonempty open subset of X contained in U such that $V\cdot W\subset U$. For every $g\in V$, $U\cap gU$ is nonempty.

Let $g\in G$ be such that $B\cap gB$ is empty. The relations

imply that $U\cap gU$ is meagre in X. Since this is an open subset and X is a Baire space, it is empty. Hence $g\notin V$, which proves the lemma.