Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open

Theorem
Let $G$ be a group acting by homeomorphisms on a topological space $X$.

Then the projection map $\pi: X \to X / G$ is open.

Proof
Let $U \subset X$ be open.

We have to show that $\pi \left({U}\right)$ is open.

By definition of quotient topology, this is the case $\pi^{-1} \left({\pi \left({U}\right)}\right)$ is open.

By definition of saturation under group action:
 * $\displaystyle \pi^{-1} \left({\pi \left({U}\right)}\right) = \bigcup_{g \mathop \in G} g U$

Because $G$ acts by homeomorphisms, $\pi^{-1} \left({\pi \left({U}\right)}\right)$ is open.

Also see

 * Projection of Subset is Open iff Saturation is Open