Continuous group action

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

Let $G$ be a topological group and $X$ a topological space. A group action $\alpha:G\times X\to X$ is continuous when $\alpha$ is a continuous map for the product topology on $G\times X$ .

Consequences

Continuity here is joint continuity on the product. Every fixed $g\in G$ then acts by a homeomorphism, with inverse given by $g^{-1}$ . Continuity is separate from faithfulness: continuity is topological regularity, while faithfulness says that only the identity acts trivially everywhere.