I was reading a proof in invariant theory, they used the fact that the action of a compact group $G$ ; namely, $\phi: G \times M \rightarrow M$, is a closed map.
I don't see this such obvious. I found a proof of this fact in Bredon's book Introduction to Compact Transformation Groups which uses nets to characterize closed sets. I am a bit unfamiliar with this notion so I'm wondering if it possible to prove this using other way.
I tried to prove that $\phi(C)^c$ is open in $M$ assuming that $C$ is closed in $G \times X$, but I don't know from where exactly really start.
Hint: The action map $\phi: G\times M\to M, (g,x)\mapsto g.m$ is isomorphic to the projection $\pi: G\times M\to M$, i.e. there is an isomorphism $\varphi: G\times M\to G\times M$ such that $\pi\varphi=\phi$. You'll find it if write spell out the last equation. Given that, you're reduced to showing that the projection $G\times M\to M$ is closed, which is a general fact from point-set topology not related to group actions (search for tube lemma if you get stuck).