Presheaf of sets on a topological space

A presheaf of sets on a topological space $X$ is a contravariant functor from the inclusion-ordered category $\mathcal{O}(X)$ of open subsets to $\mathbf{Set}$. Its maps $F(U)\to F(V)$ for $V\subseteq U$ are restriction maps.

A sheaf of sets is a presheaf in which compatible sections on any open cover glue to one unique section on the union.