## Tag `00A2`

Chapter 6: Sheaves on Spaces > Section 6.31: Open immersions and (pre)sheaves

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 $j_{p!}\mathcal{F}$.

The code snippet corresponding to this tag is a part of the file `sheaves.tex` and is located in lines 4452–4474 (see updates for more information).

```
\begin{definition}
\label{definition-j-shriek}
Let $X$ be a topological space.
Let $j : U \to X$ be the inclusion of an open subset.
\begin{enumerate}
\item Let $\mathcal{F}$ be a presheaf of sets on $U$. We define
the {\it 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.
\item Let $\mathcal{F}$ be a sheaf of sets on $U$. We define
the {\it extension of $\mathcal{F}$ by the empty set $j_!\mathcal{F}$}
to be the sheafification of the presheaf $j_{p!}\mathcal{F}$.
\end{enumerate}
\end{definition}
```

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