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