PlanetPhysics/Topological G Space

Essential Data

Let us recall the definition of a [[../TrivialGroupoid/|topological group]]; this is a [[../TrivialGroupoid/|group]] $(G,.,e)$ together with a topology on $G$ such that $(x,y)\mapsto x{y}^{-1}$ 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 .