# A corollary of proper action?

I seem to recall reading this somewhere:

If a Lie group $G$ acts on a topological space $X$ properly, then every compact subset of the orbit space $X/G$ arises as the image of a compact subset of $X$ under the projection map.

Could anyone confirm if this is true/false and why? Cheers.

Edit: by proper I mean that $G\times X\rightarrow X\times X$ given by $(g,x)\mapsto (x,g\cdot x)$ is proper.

• What is your definition of a proper action? (There are several inequivalent notions in the literature.) – Moishe Kohan Jun 14 '17 at 13:30
• If you are willing to assume local compactness of $X$ then you do not even need properness of the action and you do not need the assumption that $G$ is a Lie group. – Moishe Kohan Jun 14 '17 at 13:48
• Thanks, do you mind elaborating? Cheers. – ougoah Jun 14 '17 at 18:08

First note that if $G\times X\to G$ is a topological group action then the quotient map $\pi: X\to X/G$ is open. Now, if $X$ is locally compact (meaning that every point $x$ admits a relatively compact open neighborhood $U_x$ - one can ask for a basis of such neighborhoods but I do not need one), then $\{\pi(U_x): x\in X\}$ is an open cover of $X/G$. By compactness of $X/G$, there are finitely many points $x_1,...,x_n\in X$ such that $$X/G= \bigcup_{i} \pi(U_{x_i}).$$ Then the compact $$K= \bigcup_{i} \overline{U_{x_i}}\subset X$$ projects onto $X/G$. qed
What happens without local compactness assumption (but assuming proper discontinuity of the $G$-action, in some form) I do not know. It is a good question though.