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 $U$.

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