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 $\pi(U)$ is open.

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

By Saturation of Set Under Group Action, $\displaystyle \pi^{-1}(\pi(U)) = \bigcup_{g \in G} gU$.

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