Continuous group actions and transitivity

Question

Let $G$ be a topological group and $M$ a topological space. Suppose that $$\circ: M\times G\longrightarrow M$$ is a continuous group action of $G$ on $M$. If $\circ$ is transitive, then we can take an arbitrary $m\in M$ and the set $\mathfrak{H}$ of all cosets of $C_G(m)$ in $G$ is in a natural bijection $$\varphi:\mathfrak{H}\longrightarrow M$$ with $M$. I've been said that, in this case, $M$ is even homeomorphic to $\mathfrak{H}$ with the natural topology inherited by $G$ (an open set here is such if and only if the union of the cosets is open in $G$). I can prove that $\varphi$ is continuous composing the maps $$G\longrightarrow M\times G\longrightarrow M$$ but I cannot prove that $\varphi$ is also open. Any ideas?

Answer 1

This is not true unless you impose further assumptions: both spaces should be locally compact and Hausdorff and $G$ should be second countable. You can find a proof of this fact in the online notes by Paul Garret about 2-Solenoids. See also this related question (where I found the link to the previous notes when I had a similar doubt myself).