Example where Čech and derived functor cohomologies don't agree.

I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces.

Is there a simple example of a topological space $X$ with a sheaf $\mathcal F$ such that these two cohomologies don't agree? (I don't have any knowledge of schemes, I want $X$ to be a topological space)

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A scheme is in particular a topological space, and their sheaf cohomology only depends on the topology. In fact, they are very simple as topological spaces. :p – Zhen Lin Feb 20 '13 at 22:14
More seriously, one can prove without any hypotheses at all that Čech cohomology (after taking the direct limit over all open covers) always computes the correct $H^1$, so any counterexample has to be in $H^2$ or higher. – Zhen Lin Feb 20 '13 at 22:22
So, is there such an example for $H^2$? – Henry Wensleydale Feb 20 '13 at 22:35
Answered on MO. – Zhen Lin Feb 21 '13 at 8:22

So as to answer this question, I will give a link to a paper:

Abstract: "We construct a non-paracompact Hausdorff space for which Cech cohomology does not coincide with sheaf cohomology. Moreover, the sheaf of continuous real-valued functions is neither soft nor acyclic, and our space admits non-numerable principal bundles."

The construction is quite long, and so I'll refer you to that paper.

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mathoverflow.net/questions/122478/… You're an active MO user. Didn't you see this was already answered there, with the same reference? – Pedro Tamaroff Jun 10 '14 at 3:38
@PedroTamaroff Forgive me; didn't see that. I made the answer CW. Again, I apologize for not seeing the answer at MO. – Sanath K. Devalapurkar Jun 10 '14 at 3:51