Definition 1

A presheaf on B, relative to a basis $\mathscr{B}$ of the topology of B, is a projective system of sets, relative to the set of indices $\mathscr{B}$ ordered by the inclusion relation.

In other words (E, III, p. 52), a presheaf $\mathscr{F}$ on B relative to $\mathscr{B}$ is a pair $((\mathscr{F}(U))_{U\in\mathscr{B}},(f_{UV}))$, which is also denoted by $(\mathscr{F}(U), f_{UV})$, where $((\mathscr{F}(U))_{U\in\mathscr{B}}$ is a family of sets having $\mathscr{B}$ as set of indices and where for each pair $(U,V)$ of elements of $\mathscr{B}$ such that $U\subset V,f_{UV}$ is a mapping of $\mathscr{F}(V)$ into $\mathscr{F}(U)$, these mappings satisfying the following conditions:

(PF$_1$) The relations $U\subset V\subset W$ imply $f_{UW}=f_{UV}\circ f_{VW}$;

(PF$_2$) For every open set U $\in \mathscr{B},f_{UU}$ is the identity mapping of

A presheaf on B relative to the set of open subsets of B is simply called a presheaf on B.

Let $\mathscr{F}= (\mathscr{F}(U), f_{UV})$ be a presheaf on B, relative to a basis $\mathscr{B}$ of the topology of B. Let U be an element of the basis $\mathscr{B}$. The elements of $\mathscr{F}(U)$ are called the sections of $\mathscr{F}$ over U. If V is an element of the basis $\mathscr{B}$ containing U and $s$ an element of $\mathscr{F}(V)$, the element $f_{UV}(s)$ of $\mathscr{F}(U)$ is called the restriction of $s$ to U. If there is no danger of ambiguity concerning the mappings $f_{UV}$, the restriction of $s$ to U will be denoted by $s|U$.

Let $B'$ be an open subset of B and $\mathscr{B}'$ a basis of the topology of $B'$ such that $\mathscr{B}'\subset \mathscr{B}$. The presheaf on $B'$, relative to $\mathscr{B}'$, deduced from $\mathscr{F}$ by restriction, is called the restriction of $\mathscr{F}$ to $\mathscr{B}'$ and denoted by $\mathscr{F}|\mathscr{B}'$, the projective system $((\mathscr{F}(U))_{U\in\mathscr{B}'},(f_{UV}))$ deduced from $\mathscr{F}$ by restriction to $\mathscr{B}'$ of the set of indices (loc. cit.). When $\mathscr{F}$ is a presheaf on B and $\mathscr{B}'$ is the set of open subsets of $B'$, the presheaf $\mathscr{F}|\mathscr{B}'$ is also denoted by $\mathscr{F}|B'$ and called the presheaf deduced from $\mathscr{F}$ by restriction to $B'$.