The Stacks project

Proof. Assume $E$ is a quasi-compact open of $X$. By Lemma 5.14.2 we can write $E = p_ i^{-1}(U_ i)$ for some $i$ and some open $U_ i \subset X_ i$. Write $U_ i = \bigcup U_{i, \alpha }$ as a union of quasi-compact opens. As $E$ is quasi-compact we can find $\alpha _1, \ldots , \alpha _ n$ such that $E = p_ i^{-1}(U_{i, \alpha _1} \cup \ldots \cup U_{i, \alpha _ n})$. Hence $E_ i = U_{i, \alpha _1} \cup \ldots \cup U_{i, \alpha _ n}$ works.

Assume $E$ is a constructible closed subset. Then $E^ c$ is quasi-compact open. So $E^ c = p_ i^{-1}(F_ i)$ for some $i$ and quasi-compact open $F_ i \subset X_ i$ by the result of the previous paragraph. Then $E = p_ i^{-1}(F_ i^ c)$ as desired.

If $E$ is general we can write $E = \bigcup _{l = 1, \ldots , n} U_ l \cap Z_ l$ with $U_ l$ constructible open and $Z_ l$ constructible closed. By the result of the previous paragraphs we may write $U_ l = p_{i_ l}^{-1}(U_{l, i_ l})$ and $Z_ l = p_{j_ l}^{-1}(Z_{l, j_ l})$ with $U_{l, i_ l} \subset X_{i_ l}$ constructible open and $Z_{l, j_ l} \subset X_{j_ l}$ constructible closed. As $\mathcal{I}$ is cofiltered we may choose an object $k$ of $\mathcal{I}$ and morphism $a_ l : k \to i_ l$ and $b_ l : k \to j_ l$. Then taking $E_ k = \bigcup _{l = 1, \ldots , n} f_{a_ l}^{-1}(U_{l, i_ l}) \cap f_{b_ l}^{-1}(Z_{l, j_ l})$ we obtain a constructible subset of $X_ k$ whose inverse image in $X$ is $E$. $\square$