Sheaf - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

Let $X$ be a topological space. A sheaf $\mathcal F$ on $X$ assigns an object $\mathcal F(U)$ to every open set $U\subseteq X$ , together with restriction maps

\mathcal F(U)\longrightarrow \mathcal F(V)\qquad (V\subseteq U),

such that restrictions compose as expected. It must also satisfy two local-to-global conditions. If $U=\bigcup_i U_i$ , then sections over $U$ are determined by their restrictions to the $U_i$ ; and compatible sections $s_i\in\mathcal F(U_i)$ glue to a unique section $s\in\mathcal F(U)$ .

Examples

The elements of $\mathcal F(U)$ are called sections over $U$ . For example, continuous real-valued functions form a sheaf: functions defined on an open cover and agreeing on overlaps glue uniquely. A structure sheaf similarly records algebraic functions, while a stalk records their germs at one point.