Continuous Group Action is by Homeomorphisms
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Theorem
Let $G$ be a topological group acting continuously on a topological space $X$.
Then $G$ acts by homeomorphisms.
Proof
Let $\phi:G\times X\to X$ denote the group action.
Let $g\in G$.
The map $\phi_g : X \to X : x\mapsto \phi(g,x)$ is continuous because $\phi$ is.
Its inverse is given by $\phi_{g^{-1}}$, which is continuous as well.
Thus $\phi_g$ is an homeomorphism of $X$.
$\blacksquare$