Suppose $X$ is a Hausdorff space, and $G$ is a subgroup of homeomorphisms. We take $Y$ to be the quotient space induced by the equivalence relation;
$p \equiv q$ iff $ \exists g \in G : g(p) = q$
Also, $\pi : X \rightarrow Y$ is the mapping that sends every element to its equivalence class.
I am supposed to prove that $\pi$ is an open map.
Alright, I am not even sure that the Hausdorff condition is useful here, could be useful for the next questions in this exercise. I tried applying the definition of openness in $Y$, as a quotient space, but that does not seem to work since for an open subset $U$ of $X$, we have that $U \subseteq \pi^{-1}(\pi(U))$ but the inverse does not hold generally, I am sure of that - (for example, what if $G$ acts discontinuously on $X$). I thought about using the neighborhood definition of openness, but that also seems far fetched. So, I am stuck, and here I am.
