The Stacks project

Proof. Let $E = \bigcup _{i = 1, \ldots , n} U_ i \cap V_ i^ c$ with $U_ i, V_ i$ retrocompact open in $X$. Let $W \subset X$ be quasi-compact open. Then $E \cap W = \bigcup _{i = 1, \ldots , n} U_ i \cap V_ i^ c \cap W$. Thus it suffices to show that $U \cap V^ c \cap W$ is quasi-compact if $U, V$ are retrocompact open and $W$ is quasi-compact open. This is true because $U \cap V^ c \cap W$ is a closed subset of the quasi-compact $U \cap W$ so Lemma 5.12.3 applies. $\square$