Definition 2

Let $\mathscr{F}= (\mathscr{F}(U), f_{UV})$ be a presheaf on B. One says that $\mathscr{F}$ is a sheaf on B if, for every open subset U of B and every family $(U_i)_{i\in I}$ of open subsets of B, with union U, the following properties are satisfied:

$(F_1)$ The mapping $(f_{U_iU})_{i\in I}:\mathscr{F}(U)\rightarrow \prod_{i\in I}\mathscr{F}(U_i)$ is injective ;

$(F_2)$ For every family $(s_i)\in \prod_{i\in I}\mathscr{F}(U_i)$ such that $f_{(U_i\cap U_j)U_i}(s_i) =$

$f_{(U_i\cap U_j)U_j}(s_j)$ for every pair $(i, j)\in I\times I$, there exists an element $s$

of $\mathscr{F}(U)$ such that for every $i\in I$, one has $f_{U_iU}(s) =s_i$.