Assume that $G$ is a Polish group continuously acting on a Polish space $X$. Let $x \in X$ be a point such that $G \cdot x$, the orbit of $x$, is non-meager in its relative topology. I would like to know why this implies that $G \cdot x$ is a Baire space (as a topological subspace of $X$). I'm reading a proof of the Effros theorem and the author seems to be using this observation.
Note: We say that a topological space $X$ is Baire iff every non-empty open subset of $X$ is non-meager in $X$.
Suppose a nonempty open subset $U\subseteq Gx$ is meager in $Gx$. We may assume $x\in U$ (otherwise, rename $x$ to be an element of $U$). Let $V=\{g\in G:gx\in U\}$; then $V$ is an open neighborhood of $1$ in $G$. Now if $\{g_n:n\in\mathbb{N}\}$ is a countable dense subset of $G$, the translates $g_nV$ cover all of $G$ (proof: for any $g\in G$, there is some $g_n$ in the open set $gV^{-1}$ which means $g\in g_nV$). It follows that $Gx$ is covered by the sets $(g_nV)x=g_nU$. But since $U$ is meager, each $g_nU$ is also meager, so $Gx$ is covered by countably many meager sets and is meager.
