The question Consider a topological space $X$ and a family of sheaves (of abelian groups, say) $\; \mathcal F_i \;(i\in I)$ on $X$. Is it true that $$H^*(X,\prod \limits_{i \in I} \mathcal F_i)=\prod \limits_{i \in I} H^*(X,\mathcal F_i) \;?$$ According to Godement's and to Bredon's monographs this is correct if the family of sheaves is locally finite (In particular if $I$ is finite). [Bredon also mentions in an exercise that equality holds for spaces in which every point has a smallest open neighbourhood.]
What about the general case?
A variant Same question for $\check{C}$ech cohomology: is it true that $$\check{H}^*(X,\prod \limits_{i \in I} \mathcal F_i)=\prod \limits_{i \in I} \check{H}^*(X,\mathcal F_i) \;?$$ (Of course, $\check{C}$ech cohomology often coincides with derived functor cohomology but still the question should be considered independently)
A prayer Godement's book Topologie algébrique et théorie des faisceaux was published in 1960 and is still, with Bredon's, the most complete book on the subject. I certainly appreciate the privilege of working in a field where a book released half a century ago is still relevant: programmers and molecular biologists are not so lucky. Still I feel that a new treatise is due, in which naïve/foundational questions like the above would be addressed, and which would take the research and shifts in emphasis of half a century into account: one book on sheaf theory every 50 years does not seem an unreasonable frequency. So might I humbly suggest to one or several of the awesome specialists on MathOverflow to write one? I am sure I'm not the only participant here whose eternal gratitude they would earn.
