Exercise concerning locally constant presheaves [closed]

Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is locally constant if there exists an open cover $\mathcal{U}$ of $X$ (i.e. $X=\bigcup_{U\in\mathcal{U}}\ U$ and the elements of $\mathcal{U}$ are open) such that, for all $U\in\mathcal{U}$ and for all $P\in U$, we have $\mathscr{F}(U)=\mathscr{F}_P$.

The task is to prove that for every $U\in\mathcal{U}$ and every connected subset $V\subseteq U$, the composite

$\mathscr{F}(U) \xrightarrow{\ \text{restriction}\ } \mathscr{F}(V) \xrightarrow{\ \text{sheafification}\ } \mathscr{F}^+(V)$

is an isomorphism. This is an exercise in the Book on Algebraic Topology by Spanier. A fellow student asked me about it because he needs the result, but we were unable to figure it out.

closed as too localized by S. Carnahan♦May 10 '11 at 22:48

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• I don't think that your recent questions are appropriate for MO. Or at least you should put some effort to solve these questions by your own and show your ideas ... – Martin Brandenburg May 10 '11 at 22:44
• Perhaps you should try asking on math.stackexchange.com – S. Carnahan May 10 '11 at 22:48
• I'm sorry, I had read the FAQ and I am not quite sure how my questions are inappropriate. In particular, how is this particular question any less demanding or interesting than, say, that one: mathoverflow.net/questions/24361/… – Jesko Hüttenhain May 10 '11 at 23:19
• The question you linked, while easy, is interesting because it is about local-versus-global phenomena. Your question is a diagram chase. – S. Carnahan May 11 '11 at 4:49
• Note also that saying "$\mathcal{F}(U) = \mathcal{F}_P$" seems slightly sloppy to me: what you mean is that the canonical map from the guy on the left to the guy on the right is an isomorphism. – Pete L. Clark May 11 '11 at 5:02