topological G-space

0.1 Essential data

Let us recall the definition of a topological group; this is a group $(G,.,e)$ togetherwith 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 0.1.

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 $X$ iscontinuous if $a:G\times X\to X$ is continuous; with these conditions, $X$ is defined to bea topological G-space.