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In chapter I, appendix to section 2 of the book "abelian varieties" by Mumford, we consider a discrete group $G$ acting freely and discontinuously on a good topological space $X$ (i.e., $\forall x \in X$, $x$ has a nbhd $U_x$ such that $U_x\cap \sigma(U_x)=\emptyset$, all $\sigma \in G$, $\sigma\neq e$). Let $Y$ be the orbit space $X/G$, and $\pi: X\to Y$ the projection.

Then it was claimed that we may choose a covering $\{V_i\}_{i\in I}$ of $Y$ such that for each $i$,

  • $\pi^{-1}(V_i)=\cup_{\sigma\in G} \sigma U_i$, $U_i\subset X$ open such that $\pi$ restricted to $U_i$ is a homeomorphism.

  • $\forall i, j$ there exists at most one $\sigma\in G$ such that $U_i\cap \sigma U_j \neq \emptyset$.

I wonder why such a covering always exists and how one may construct them. If $Y$ is a smooth manifold, then I believe taking "convex" $V_i$ will do the trick. But I am having trouble to come up with a purely topological method in the general situation (one might have to specify what Mumford means by "good topological space").

Thank you.

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    As a side remark, I just realize that the section in Mumford's book that I was referring to is just a special case of Serre-Hochschild spectral sequence.2011-06-15

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