I'm trying to understand the footnote to Example 5.3 in Wiegand - Sheaf cohomology of locally compact totally disconnected spaces which is about constructing a locally compact Hausdorff and totally disconnected space whose sheaf cohomology with constant sheaf $\mathbb Z/2\mathbb Z$ coefficients doesn't vanish. An explicit covering is given by compact open sets with no three intersecting and the sheaf cohomology is shown to coincide with the Čech cohomology over any covering by compact open sets. So things boil down to computing the Čech cohomology in this sheaf with respect to this covering.

In the end, since we are dealing essentially with boolean algebras, it turns into the following problem which I don't immediately see how to solve. If $Y$ is a set, let $B(Y)$ be the boolean algebra of finite and cofinite subsets of $Y$. Let $S$ be a countably infinite set and $T$ an uncountable set. Define a mapping $\Phi\colon B(T)^S\times B(S)^T\to 2^{S\times T}$ as follows. If $f\colon S\to B(T)$ and $g\colon T\to B(S)$ are maps, send $(f,g)$ to the subset of $S\times T$ consisting of all pairs $(s,t)$ with $s\in g(t)$ or $t\in f(s)$, but not both. Then I think the footnote is equivalent to the claim that $\Phi$ is not onto.

**Question**: why is $\Phi$ not onto?

specificpair of sets? If so, then the obvious thing seems to be to try to take $T = 2^S$ and to see if $\Phi$ hits the graph of $\in$. $\endgroup$ – LSpice Jan 18 at 14:59