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Let $\mathcal F_i, i\in I$, be a directed family of sheaves of abelian groups on a paracompact Hausdorff space $X$. Let $\mathcal F=\varinjlim F_i$ denote the direct limit sheaf. Is it true that $\mathcal F(X)=\varinjlim \mathcal F_i(X)$? If not, is it true under extra assumptions on $\mathcal F_i$'s?

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In general not: take $X$ to be $\mathbb{Z}_{\geq 0}$ and $\mathcal{F}_i$ the constant sheaf supported at $X\cap [0,i]$. The direct limit will be the constant sheaf on $X$ and will have many more sections than there are in the direct limit of the sections of the $\mathcal{F}_i$'s.

However, if you consider compactly supported cohomology (or if $X$ itself is compact), then everything is fine. See e.g. Godement, theorem 4.12.1.

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