Lemma 5.24.1. Let \mathcal{I} be a category. Let i \mapsto X_ i be a diagram of spectral spaces such that for a : j \to i in \mathcal{I} the corresponding map f_ a : X_ j \to X_ i is spectral.
Given subsets Z_ i \subset X_ i closed in the constructible topology with f_ a(Z_ j) \subset Z_ i for all a : j \to i in \mathcal{I}, then \mathop{\mathrm{lim}}\nolimits Z_ i is quasi-compact.
The space X = \mathop{\mathrm{lim}}\nolimits X_ i is quasi-compact.
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