TITLE: Example where Čech and derived functor cohomologies don't agree.
QUESTION [7 upvotes]: 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)

REPLY [2 votes]: So as to answer this question, I will give a link to a paper:

http://arxiv.org/pdf/1309.2524v1.pdf.
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.