The Stacks project

Lemma 6.30.4. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Assume that for every triple $U, U', U'' \in \mathcal{B}$ with $U' \subset U$ and $U'' \subset U$ we have $U' \cap U'' \in \mathcal{B}$. For each $U \in \mathcal{B}$, let $C(U) \subset \text{Cov}_\mathcal {B}(U)$ be a cofinal system. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{B}$. The following are equivalent

  1. The presheaf $\mathcal{F}$ is a sheaf on $\mathcal{B}$.

  2. For every $U \in \mathcal{B}$ and every covering $\mathcal{U} : U = \bigcup U_ i$ in $C(U)$ and for every family of sections $s_ i \in \mathcal{F}(U_ i)$ such that $s_ i|_{U_ i \cap U_ j} = s_ j|_{U_ i \cap U_ j}$ there exists a unique section $s \in \mathcal{F}(U)$ which restricts to $s_ i$ on $U_ i$.

Proof. This is a reformulation of Lemma 6.30.3 above in the special case where the coverings $\mathcal{U}_{ij}$ each consist of a single element. But also this case is much easier and is an easy exercise to do directly. $\square$

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