In Hartshorne, p.61, definition of a presheaf. The condition (3) reads:
if U is an open set, if {$V_i$} is an open covering of U, and if $s \in \mathscr{F}(U)$ is an element of such that $s|_{V_i} =0$ for all $i$ then $s =0$.
Since the restriction can only be defined if $V_{i} \subseteq U$ for each $i$, should’nt there be an additional condition on the cover, for example the union of subsets in the cover must be equal to $U$ ?