Definition 6.7.1. Let $X$ be a topological space.

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*sheaf $\mathcal{F}$ of sets on $X$*is a presheaf of sets which satisfies the following additional property: Given any open covering $U = \bigcup _{i \in I} U_ i$ and any collection of sections $s_ i \in \mathcal{F}(U_ i)$, $i \in I$ such that $\forall i, j\in I$\[ s_ i|_{U_ i \cap U_ j} = s_ j|_{U_ i \cap U_ j} \]there exists a unique section $s \in \mathcal{F}(U)$ such that $s_ i = s|_{U_ i}$ for all $i \in I$.

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*morphism of sheaves of sets*is simply a morphism of presheaves of sets.The category of sheaves of sets on $X$ is denoted $\mathop{\mathit{Sh}}\nolimits (X)$.

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