Definition 6.31.3. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

1. Let $\mathcal{F}$ be a presheaf of sets on $U$. We define the extension of $\mathcal{F}$ by the empty set $j_{p!}\mathcal{F}$ to be the presheaf of sets on $X$ defined by the rule

$j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} \emptyset & \text{if} & V \not\subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right.$

with obvious restriction mappings.

2. Let $\mathcal{F}$ be a sheaf of sets on $U$. We define the extension of $\mathcal{F}$ by the empty set $j_!\mathcal{F}$ to be the sheafification of the presheaf $j_{p!}\mathcal{F}$.

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