The Stacks project

Proof. Apply Lemma 35.13.1 to the map $\coprod _{i \in I} T_ i \to T$. It implies there exists a subset $W \subset T$ such that $W_ i = f_ i^{-1}(W)$ for each $i$, namely $W = \bigcup f_ i(W_ i)$. To see that $W$ is open we may work Zariski locally on $T$. Hence we may assume that $T$ is affine. Using Topologies, Definition 34.9.1 we may choose a standard fpqc covering $\{ g_ j : V_ j \to T\} _{j \in J}$ which refines $\{ T_ i \to T\} _{i \in I}$. Let $\alpha : J \to I$ and $h_ j : V_ j \to T_{\alpha (j)}$ be as in Sites, Definition 7.8.1. Then $g_ j^{-1}(W) = h_ j^{-1}(W_{\alpha (j)})$. Thus we may assume $\{ f_ i : T_ i \to T\} $ is a standard fpqc covering. In this case we may apply Morphisms, Lemma 29.25.12 to the morphism $\coprod T_ i \to T$ to conclude that $W$ is open. $\square$