# Does the orbit map induce an isomorphism (or a monomorphism) in Alexander -Spanier cohomology?

Let $G$ be a compact and totally disconnected group acting on a paracompact space $X$.

Does the orbit map $X \rightarrow X/G$ induce an isomorphism (or a monomorphism) in Alexander-Spanier cohomology with closed support?

Theorem. Let $G$ be a totally disconnected compact group that acts on a locally compact Hausdorff space $X$, and let $k$ be a field of characteristic $0$. Then the orbit projection $X\rightarrow X/G$ induces an isomorphism $$H_c^{*}(X/G;k)\cong \text{Fix}(G;H_c^{*}(X;k)).$$