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(I feel like the following question is probably something really basic. Oh, well.)

Recall that a sheaf $F$ on a topological space $X$ is flasque if for every open subset $U\subseteq X$, the restriction map $F(X)\to F(U)$ is surjective. Since this definition doesn't use the sheaf axioms, we can also use it to define flasqueness for presheaves.

My question is then: If $F$ is a flasque presheaf, is its sheafification $F^+$ also flasque?

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    $\begingroup$ No, take for example the constant presheaf (obviously flasque), its sheafification is the constant sheaf which is not flasque. $\endgroup$ – Roland Aug 3 '17 at 21:00
  • $\begingroup$ @Roland See! I knew it was basic. :P You should add that as an answer. $\endgroup$ – Avi Steiner Aug 3 '17 at 21:01
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No, take for example the constant presheaf (obviously flasque), its sheafification is the constant sheaf which is not flasque in general (and in fact has very interesting cohomology).

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