Exercise 111.32.9. Let $X$ be a topological space. Suppose we are given a collection of abelian groups $A_ x$ indexed by $x \in X$. Show that the rule $U \mapsto \prod _{x \in U} A_ x$ with obvious restriction mappings defines a sheaf $\mathcal{G}$ of abelian groups. Show, by an example, that usually it is not the case that $\mathcal{G}_ x = A_ x$ for $x \in X$.
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