\begin{definition}[Definition:Group Action by Homeomorphisms]
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''' {{iff}} for all $g \in G$, the mapping:
:$\phi_g : X \to X : x \mapsto \phi \left({g, x}\right)$
is a homeomorphism.
\end{definition}