# The Stacks Project

## Tag 00A1

Definition 6.31.2. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

1. Let $\mathcal{G}$ be a presheaf of sets, abelian groups or algebraic structures on $X$. The presheaf $j_p\mathcal{G}$ described in Lemma 6.31.1 is called the restriction of $\mathcal{G}$ to $U$ and denoted $\mathcal{G}|_U$.
2. Let $\mathcal{G}$ be a sheaf of sets on $X$, abelian groups or algebraic structures on $X$. The sheaf $j^{-1}\mathcal{G}$ is called the restriction of $\mathcal{G}$ to $U$ and denoted $\mathcal{G}|_U$.
3. If $(X, \mathcal{O})$ is a ringed space, then the pair $(U, \mathcal{O}|_U)$ is called the open subspace of $(X, \mathcal{O})$ associated to $U$.
4. If $\mathcal{G}$ is a presheaf of $\mathcal{O}$-modules then $\mathcal{G}|_U$ together with the multiplication map $\mathcal{O}|_U \times \mathcal{G}|_U \to \mathcal{G}|_U$ (see Lemma 6.24.6) is called the restriction of $\mathcal{G}$ to $U$.

The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4422–4443 (see updates for more information).

\begin{definition}
\label{definition-restriction}
Let $X$ be a topological space.
Let $j : U \to X$ be the inclusion of an open subset.
\begin{enumerate}
\item Let $\mathcal{G}$ be a presheaf of sets, abelian groups or
algebraic structures on $X$. The presheaf $j_p\mathcal{G}$ described
in Lemma \ref{lemma-j-pullback} is called
the {\it restriction of $\mathcal{G}$ to $U$} and denoted $\mathcal{G}|_U$.
\item Let $\mathcal{G}$ be a sheaf of sets on $X$, abelian groups or
algebraic structures on $X$. The sheaf $j^{-1}\mathcal{G}$ is called
the {\it restriction of $\mathcal{G}$ to $U$} and denoted $\mathcal{G}|_U$.
\item If $(X, \mathcal{O})$ is a ringed space, then the pair
$(U, \mathcal{O}|_U)$ is called the
{\it open subspace of $(X, \mathcal{O})$ associated to $U$}.
\item If $\mathcal{G}$ is a presheaf of $\mathcal{O}$-modules
then $\mathcal{G}|_U$ together with the multiplication map
$\mathcal{O}|_U \times \mathcal{G}|_U \to \mathcal{G}|_U$
(see Lemma \ref{lemma-pullback-module})
is called the {\it restriction of $\mathcal{G}$ to $U$}.
\end{enumerate}
\end{definition}

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