A $G$-space is (generally) a topological space $X$ equipped with a continuous action by a topological group $G$.
A $G$-space $X$ is called proper if the map $\theta:G\times X\rightarrow X\times X, ~(g,x)\mapsto (x,g\cdot x)$ is proper in the sense that the inverse images of compact sets are compact sets.
Suppose $G$ is a compact group, how to show that every $G$-space $X$ is proper?
Best Answer
We need that $X$ is Hausdorff. Let $C$ be a compact subset of $X\times X$, and $p_1:X\times X\rightarrow X$ the projection on the first factor. $p(C)=D$ is compact. $\theta^{-1}(C)$ is contained in $G\times D$ which is compact as product of compact sets. Since $C$ is closed, $\theta^{-1}(C)$ is closed, and we deduce that it is compact.