Let *X* be a compact space. 


Recall that its Čech cohomology $H^\bullet(X,\mathbb Z)$
is given by the colomit $\mathrm{colim}\_U\big(H^\*(C^\bullet(U;\mathbb Z),\delta)\big)$, where $U=(U\_i)$ runs over all open covers of *X*, ordered by refining. 
For completeness, let us also recall that the *n*-cochains $C^n(U;\mathbb Z)$ are the group of continuous ℤ-valued functions on $\bigsqcup U\_{i\_1}\cap\ldots\cap U\_{i\_{n+1}}$.
  

Since *X* is compact, we may restrict ourselves to finite covers, without modifying the answer.


•  **Definition:**  A closed cover $V=(V\_i)$ of *X* is a finite collection of closed subsets $V\_i$ whose union is *X*.


We may now consider the modified Čech cohomology $\tilde H^\bullet(X,\mathbb Z)$, where we use closed covers instead of open covers. 


•  **Question:** Are $\tilde H^\bullet(X,\mathbb Z)$ and $H^\bullet(X,\mathbb Z)$ isomorphic?


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*PS:* I know how to show that $\tilde H^1(X,\mathbb Z)$ and $H^1(X,\mathbb Z)$ are isomorphic, by using the fact that they both classify ℤ-principal bundles.



Čech cohomology of compact spaces via closed covers?
As Angelo says, there is a Mayer-Vietoris (spectral) sequence for closed covers. That comes from an exact sequence of sheaves, which also shows that closed covers are covers in the sense of a Grothendieck topology. Probably it's true for proper surjective maps in general. 


I think that means that there is a geometric morphism from the topos of closed covers to the usual topos. This yields a comparison map $H^i\_{closed}\to H^i\_{open}$. If the space is Hausdorff, one can use partitions of unity to refine open covers by closed covers. That shows that the comparison map is surjective: any cohomology class is defined on an open cover, so one can refine it to a closed cover, so it comes from the closed cohomology. If one is careful about the bookkeeping, probably one can arrange a splitting of the Čech complexes, as in André's comment. But we can do better without mentioning this other topos.


We can compute the usual cohomology using the Čech process for open covers. But we could also do it for both open covers and closed covers. This won't change the answer, since closed covers are covers for the usual Grothendieck topology. But now we can get rid of the open covers, since the closed covers are cofinal.


I'm not sure what happens in the non-Hausdorff case. It is certainly not possible to refine open covers by closed covers, but it seems to me that the line with the doubled origin works anyway.


