Definition:Compatible Family of Sections on Topological Space

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 $U \subset S$ be an open subset of $T$.

Let $\family {f_i \in \map \FF {U_i} }_{i \mathop \in I}$ be a family.

Then $\family {f_i \in \map{\FF} {U_i} }_{i \mathop \in I}$ is a compatible family of sections
 * $\forall i, j \in I : \map {\operatorname {res}_{U_i \mathop \cap U_j}^{U_i} } {f_i} = \map {\operatorname {res}_{U_i \mathop \cap U_j}^{U_j} } {f_j}$

Also see

 * Definition:Sheaf of Sets on Topological Space