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

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

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 $U_ i$.

## Comments (0)

There are also: