Lemma 2

Let G be a locally compact group countable at infinity, M a Baire space. Suppose that G operates on the left continuously and transitively in M. For every $x \in M$, let $H_x$ be the stabilizer of $x$ in G, so that the mapping $s \mapsto sx$ of G onto M defines, by passage to the quotient, a continuous bijection $\varphi_x$ of $G/H_x$ onto M. Then $\varphi_x$ is a homeomorphism of $G/H_x$ onto M (in other words (GT, III, §2, No. 5) M is a topological homogeneous space).

Let $x_0 \in M$. It suffices to prove (loc. cit., Prop. 15) that the mapping $s \mapsto sx_0$ transforms every neighborhood V of e in G into a neighborhood of $x_0$ in M. Let W be a symmetric compact neighborhood of e such that $W^2 \subset V$. By hypothesis, G is the union of a sequence of compact sets, hence of a sequence $(s_n W)$ of translates of W. Then M is the union of the sequence of compact sets $(s_n Wx_0)$. Since M is a Baire space, there exists an index n such that $s_n Wx_0$ has an interior point $s_n wx_0$ (where $w \in W$). Consequently $x_0$ is an interior point of

so that $Vx_0$ is a neighborhood of $x_0$ in M.