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It is well-known the next theorem at Chapter III, Theorem 7.2. in Bredon's Introduction to compact transformation groups book.

Theorem: Let $X$ be a paracompact $G$-space with $G$ finite and let $\pi:X \rightarrow X/G$ be the orbit map. If $k$ is a field of characteristic zero, then $$\pi^*:H^*(X/G;k) \rightarrow H^*(X;k)^G$$ is an isomorphism, where $H^*(X;k)^G$ is the fixed point set of $G$ action on $H^*(X;k)$.

My Question: If $G$ is a compact, totally disconnected group and $X$ is a paracompact $G$-space, then $$\pi^*:H^*(X/G;k) \rightarrow H^*(X;k)^G$$ is an isomorphism?

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    $\begingroup$ I don't know the answer to your question, but it might depend on what cohomology theory you're using. Note that Bredon's statement is for Cech cohomology (and for Bredon, paracompact is defined to include the Hausdorff separation axiom). If $X$ and $X/G$ are locally contractible, then Cech and singular cohomology agree. You can see a nice summary of the relationships between various cohomology theories in the introduction to Sella's paper arxiv.org/abs/1602.06674 $\endgroup$ – Dan Ramras Jul 17 '18 at 14:23
  • $\begingroup$ I use Cech cohomology theory too. $\endgroup$ – Mehmet Onat Jul 17 '18 at 20:22

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