Lemma 70.5.7. Notation and assumptions as in Situation 70.5.5. For any quasi-compact open subspace $U \subset X$ there exists an $i$ and a quasi-compact open $U_ i \subset X_ i$ whose inverse image in $X$ is $U$.

**Proof.**
Follows formally from the construction of limits in Lemma 70.4.1 and the corresponding result for schemes: Limits, Lemma 32.4.11.
$\square$

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