Definition:Sheaf of Sets on Topological Space/Definition 1

Definition 1
Let $T = \struct {S, \tau}$ be a topological space.

Let $\FF : \map {\mathbf {Ouv} } T^{\mathrm {op} } \to \mathbf {Set}$ be a presheaf of sets on $T$.

$\FF$ is a sheaf of sets on $T$, it satisfies the following gluing property:

For any open subset $U \subset S$ of $T$ and
 * for any open cover $\family {U_i}_{i \mathop \in I}$ of $U$
 * and for any compatible family of sections $\family {f_i}_{i \mathop \in I}$ with $f_i \in \map\FF{U_i}$ for $i \mathop \in I$
 * there exists a unique $f \mathop \in \map \FF U$, such that
 * $\forall i \in I: \map {\operatorname {res}_{U_i}^U} f = f_i$

Also see

 * Equivalence of Definitions of Sheaf of Sets
 * Sheaf of Sets iff Set-Valued Sheaf
 * Definition:Sheaf on Topological Space