A <Zariski cover> of a <scheme> $T$ is a family of <open immersions> $(j_i:U_i\hookrightarrow T)_{i\in I}$ whose images cover its underlying <topological space>:
$$
\boxed{T=\bigcup_{i\in I}j_i(U_i).}
$$
There is no finiteness requirement. Intersections used in the <sheaf on a site> condition are the <pullbacks> $U_i\times_TU_j$, identified with the corresponding <open subschemes> $U_i\cap U_j$.

Solution

Past exam of the introduction to schemes course of the University of Oxford / 2023 / 1 / a / ii / Solution

 Solution (source code)