Local Property of Flasque Sheaves

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Let ##X## be a topological space, and let ##\mathscr{F}## be a sheaf on ##X##. Show that if ##\mathscr{U}## is an open cover of ##X## such that the restriction ##\mathscr{F}|_U## is flasque for every ##U\in \mathscr{U}##, then ##\mathscr{F}## is flasque.

Note: A sheaf ##\mathscr{G}## on ##X## is flasque if for all open subsets ##U\subset X##, the restriction map ##\mathscr{G}(X) \to \mathscr{G}(U)## is surjective.

 

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For this problem you will need to use Zorn's lemma.

 

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Spoiler

Fix an open subset ##U_0\subset X## and an ##s\in \mathscr{F}(U_0)##. Let ##\Sigma## be the collection of all pairs ##(W,t)##, where ##W## is open in ##X## with ##W\supset U_0## and ##t\in \mathscr{F}(W)## such that ##t|_{U_0} = s##. Partially order ##\Sigma## by declaring ##(W,t) \le (W',t')## if ##U_0 \subset W \subset W'## and ##t'|_{W} = t##. Then ##\Sigma## is a nonempty inductive set, and by Zorn's lemma there is a maximal element ##(V,r)## of ##\Sigma##. Suppose ##V \neq X##. There is an ##x\in X\setminus V##; let ##U\in \mathscr{U}## be an open neighborhood of ##x##. Since ##\mathscr{F}|_U## is flasque, there is an ##\alpha \in \mathscr{F}(U)## such that ##\alpha|_{U\cap V} = r|_{U\cap V}##. The sheaf property produces a ##\beta\in \mathscr{F}(U\cap V)## such that ##\beta|_V = r##. The pair ##(U\cup V, \beta) > (V,r)## in ##\Sigma##, contradicting maximality of ##(V,r)##. Hence, ##V = X## and ##\mathscr{F}## is flasque.