Let $X$ be a topological space and let $\mathcal{F}$ be a sheaf of abelian groups on $X.$ It is clear to me that $\check{H^{0}}(X,\mathcal{F})=H^{0}(X,\mathcal{F}).$ It is also true that $\check{H^{1}}(X,\mathcal{F})=H^{1}(X,\mathcal{F})$ (Harthshorne, exe. III.4.4). But what about higher cohomology groups?
I belive that for $k\geq2$ it is false in general that $\check{H^{k}}(X,\mathcal{F})=H^{k}(X,\mathcal{F})$ but I don't know how to construct a counterexample. Is there a well known example of such $X$ and $\mathcal{F}?$ I will be thankful for any help.