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 $\phi_g : X \to X : x \mapsto \phi(g,x)$

Let $U\subset X$ be open.

By Inverse Image of Set Under Mapping from Product of Sets:
 * $\displaystyle\phi^{-1}(U) = \bigcup_{g\in G} \left( \{g\} \times \phi_g^{-1}(U) \right)$.

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

Thus $\phi$ is continuous.

Also see

 * Continuous Group Action is by Homeomorphisms