Example 1

Let B be a topological space, $\mathscr{B}$ a base of the topology of B and F a set. Take for $\mathscr{F}$ the presheaf on B relative to $\mathscr{B}$ defined by $\mathscr{F}(U) = F$ for every $U\in \mathscr{B}$ and $r_{UV}$ = Id$_F$ for every pair $(U,V)$ of elements of $\mathscr{B}$ such that $U\subset V$. The mapping $[U, s, x]\mapsto (x, s(x))$ is a B-isomorphism of the B-space $E_{\mathscr{F}}$ onto the B-space $B\times F$ where F is endowed with the discrete topology. It will be noted that when $\mathscr{B}$ is the set of open subsets of B, the presheaf $\mathscr{F}$ on B is a sheaf only if the set F is reduced to a point (cf. I, p. 43, remark).