Example 6.7.6. Let $X$ be a topological space. Suppose for each $x\in X$ we are given an abelian group $M_ x$. Consider the presheaf $\mathcal{F} : U \mapsto \bigoplus _{x \in U} M_ x$ defined in Example 6.4.5. This is not a sheaf in general. For example, if $X$ is an infinite set with the discrete topology, then the sheaf condition would imply that $\mathcal{F}(X) = \prod _{x\in X} \mathcal{F}(\{ x\} )$ but by definition we have $\mathcal{F}(X) = \bigoplus _{x \in X} M_ x = \bigoplus _{x \in X} \mathcal{F}(\{ x\} )$. And an infinite direct sum is in general different from an infinite direct product.

However, if $X$ is a topological space such that every open of $X$ is quasi-compact, then $\mathcal{F}$ is a sheaf. This is left as an exercise to the reader.

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