Discrete Group Acts Continuously iff Acts by Homeomorphisms

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

Then the following are equivalent:
 * $G$ acts continuously
 * $G$ acts by homeomorphisms

Proof
If $G$ acts continuously, then by Continuous Group Action is by Homeomorphisms, $G$ acts by homeomorphisms

Let $G$ act by homeomorphisms

Let $\phi: G \times X \to X$ denote the group action.

For $g \in G$, denote $\phi_g : X \to X : x \mapsto \map \phi {g, x}$

Let $U \subset X$ be open.

By Inverse Image of Set Under Mapping from Product of Sets:
 * $\ds \map {\phi^{-1} } U = \bigcup_{g \mathop \in G} \paren {\set g \times \map {\phi_g^{-1} } U}$

By definition of product topology, $\map {\phi^{-1} } U$ is open in $G \times X$.

Thus $\phi$ is continuous.

Also see

 * Continuous Group Action is by Homeomorphisms