# Definition:Group Action by Homeomorphisms

## Definition

Let $G$ be a group.

Let $X$ be a topological space.

Let $\phi: G \times X \to X$ be a group action

Then $G$ acts by homeomorphisms if and only if for all $g \in G$, the mapping:

$\phi_g : X \to X : x \mapsto \phi \left({g, x}\right)$

is a homeomorphism.