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.