Čech cohomology of compact spaces via closed covers? 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?

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.
 A: If $V$ is a finite closed cover of a space $X$, with the property that the higher cohomology of all finite intersections vanishes, then the Čech cohomology of $V$ coincides with that of $X$. This is because there is a Mayer-Vietoris spectral sequence for finite closed covers, completely analogous to that for open covers, which is is easy to obtain from the long exact sequence of sheaves
$$
0 \to \mathbb Z_X \to \bigoplus \mathbb Z_{V_i} \to \bigoplus \mathbb Z_{V_i \cap V_j} \to \cdots
$$
(essentially, this comes from the nerve of the cover). Here $\mathbb Z_Y$ denotes the constant sheaf on $Y$, pushed forward to $X$.
On the other hand, I would be a little nervous about what happens going to the limit, even for very nice spaces. I have never thought about it, but it seems to me that there could be finite nasty closed covers of a manifold that don't admit nice refinements; so I would deem it likely that one has to restrict the covers to a mild class (I could be completely wrong, of course).
A: 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.
