I am reading Hartshorne's book on algebraic geometry, which defines a sheaf to be a presheaf $\mathscr F$ on a topological space $X$ such that:
- For all open sets $U$ and open coverings $\{V_i\}$ of $U$ in $X$, if $s\in\mathscr F(U)$ with $\forall i.\;s\rvert_{V_i}=0$, then $s=0$;
- (more requirements)
However, the restriction maps $\cdot\,\rvert_V\colon\;\mathscr F(U)\to \mathscr F(V)$ were previously only defined if $V\subseteq U$. How does this definition handle the case that $V_i\not\subseteq U$? Does one only consider those sets $V_i$ of the covering that are fully contained in $U$?
I remember being confused about this when I first started learning sheaves. In sheaf theory, one always takes open covers by subsets for just this reason.
If it gives you some peace of mind, note that if you have an open cover $\{V_i\}$ of an open set $U$, you can always replace it by the open cover by subsets $\{U\cap V_i\}$.
Also, only considering the open sets of a cover which happen to be contained in $U$ wont work because there are many open covers which have no such sets (the whole space $X$ covers $U$ for example), yet the first bullet then would say sections on $U$ are zero.