Polish G-space

Definition 0.1.

Let $X$ be a topological $G$-space (http://planetmath.org/TopologicalGSpace), and $G$ its associated topological group,that is, such that an action $a$ of $G$ on $X$ is continuous if $a:G\times X\to X$ is continuous.If $G$ is a Polish group and $X$ is also a Polish space, then $X$ is called a Polish G-space.