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.

Then:

Since each $\set g \times {\phi _g^{-1} \sqbrk U }$ is open by, $\phi^{-1} \sqbrk U$ is open.

Thus $\phi$ is continuous.

Also see

 * Continuous Group Action is by Homeomorphisms