Continuous Group Action is by Homeomorphisms

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 $X$.

Also see

 * Discrete Group Acts Continuously iff Acts by Homeomorphisms