Definition:Sheaf of Sets on Topological Space/Definition 1
Jump to navigation
Jump to search
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 $T$, if and only if 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
- $\forall i \in I: \map {\operatorname {res}_{U_i}^U} f = f_i$
- there exists a unique $f \mathop \in \map \FF U$, such that
- and for any compatible family of sections $\family {f_i}_{i \mathop \in I}$ with $f_i \in \map\FF{U_i}$ for $i \mathop \in I$