PlanetPhysics/Topological G Space

Essential Data
Let us recall the definition of a topological group; this is a group $$(G, . ,e)$$ together with a topology on $$G$$ such that $$(x,y) \mapsto xy^{-1}$$ is continuous, i.e., from $$G \times G$$ into $$G$$. Note also that $$G \times G$$ is regarded as a topological space defined by the product topology.

Definition: Topological Group
Consider $$G$$ to be a topological group with the above notations, and also let $$X$$ be a topological space, such that an action $$a$$ of $$G$$ on $$X$$ is continuous if $$a : G \times X \to X$$ is continuous; with these conditions, $$X$$ is defined to be a topological G-space.