Remark

Let $\mathscr{F}$ be a presheaf on B. For every open set U of B, $f_{\emptyset U}$ is a map of $\mathscr{F}(U)$ into $\mathscr{F}(\emptyset )$, hence $\mathscr{F}(\emptyset )$ is not empty as soon as there exists an open set U for which $\mathscr{F}(U)$ is not empty. If $\mathscr{F}$ is a sheaf, $\mathscr{F}(\emptyset )$ is a one-element set; this is seen by applying $(F_1)$ and $(F_2)$ to the covering of the empty set by the empty family ($I =\emptyset$ ).

Let $\mathscr{F}$ be a sheaf on B and let $B'$ be an open subset of B; the presheaf $\mathscr{F}|B'$ deduced from $\mathscr{F}$ by restriction to $B'$ is a sheaf, called the sheaf deduced from $\mathscr{F}$ by restriction to $B'$.