Problems in proving that the projection on the quotient is an open map I consider $G\leq Hom(X)$, where $X$ is a topological space and $Hom(X)$ the groupof all homeomorphisms $f:X \rightarrow X$.
I have an equivalence relation on $X$:
$\;x \sim y\;$ iff $\;\exists \varphi \in G$ s.t. $\; \varphi(x)=y$.
I call $X/G$ the quotient space and I want to prove that the map $p:X \rightarrow X/G$ is open.
If $U$ is an open set of $X$ then I have that $p^{-1}(p(U)) = \cup_{g \in G} g(U)$ is open because union of open sets.
How can I say from this fact that $p$ is open? Why $p(U)$ should be open?
Thanks!

The open sets in the quotient topology induced by $p$ are the subsets $A\subseteq X/G$ such that $p^{-1}(A)$ is open set in $X$. 
So when you proved that for any open set $U\subseteq X$, $p^{-1}(p(U))$ is open, you proved that $p(U)$ is open in $X/G$. And so, you prove that $p$ is open. 
