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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