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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.

Sam95
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    Is $M$ a manifold? If so, then at some point you should be able to use the fact that the image of a compact set is compact and that any compact subset of a Hausdorff space is closed – Exit path Nov 22 '17 at 04:17

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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).

Hanno
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