Let $G$ be a topological group acting on a set $X$. Let $x \in X$ and consider the orbit $G.x$ endowed with the topology coming from the quotient $G/ Stab(x)$.
If $G^0$ is the connected component of the identity in $G$, is the $G^0$ orbit of $x$ $G^0.x$ a connected component of $G.x$ ?