Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open
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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 $\pi \sqbrk U$ is open.
By definition of quotient topology, this is the case if and only if $\pi^{-1} \sqbrk {\pi \sqbrk U}$ is open.
By definition of saturation under group action:
- $\ds \pi^{-1} \sqbrk {\pi \sqbrk U} = \bigcup_{g \mathop \in G} g U$
Because $G$ acts by homeomorphisms, $\pi^{-1} \sqbrk {\pi \sqbrk U}$ is open.
$\blacksquare$