Definition:Sheaf of Sections

Definition

Let $f : Y \to X$ be a continuous map.

Let $\FF : \map {\mathbf {Ouv} } X^{\mathrm {op} } \to \mathbf {Set}$ be the presheaf defined as follows:

For any open subset $U \subset X$, let $\map \FF U$ be the set of continuous sections of $f {\restriction_U} : \map {f^{-1} } U \to U$.

For any two open subsets $U \subset V \subset X$, define

$\operatorname {res}_V^U : \map \FF V \to \map \FF U, \quad s \mapsto s {\restriction_V}$

by the restriction $s {\restriction_V}$ of sections $s : U \to \map {f^{-1} } U$ to $V$.

$\FF$ is called the sheaf of sections of $f$.