Definition:Sheaf of Sets on Topological Space/Definition 2

Definition
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 $T$. Let $\map {\operatorname {Sp\acute e} } \FF$ be the étalé space of $\FF$.

Let $\FF'$ be the sheaf of sections of $\map {\operatorname {Sp \acute e} } \FF \to T$.

$\FF$ is a sheaf of sets on $T$ the canonical mapping $\FF \to \FF'$ is an isomorphism.

Also see

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