The Stacks project

Lemma 6.32.3. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor

\[ i_* : \textit{Ab}(Z) \longrightarrow \textit{Ab}(X) \]

is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = 0$ for all $x \in X \setminus Z$.

Proof. Omitted. $\square$

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