Lemma 5.23.14. A topological space is spectral if and only if it is a directed inverse limit of finite sober topological spaces.
Proof. One direction is given by Lemma 5.23.12. For the converse, assume X is spectral. Then we may assume X \subset \prod _{i \in I} W is a subset closed in the constructible topology where W = \{ 0, 1\} as in Lemma 5.23.13. We can write
\prod \nolimits _{i \in I} W = \mathop{\mathrm{lim}}\nolimits _{J \subset I\text{ finite }} \prod \nolimits _{j \in J} W
as a cofiltered limit. For each J, let X_ J \subset \prod _{j \in J} W be the image of X. Then we see that X = \mathop{\mathrm{lim}}\nolimits X_ J as sets because X is closed in the product with the constructible topology (detail omitted). A formal argument (omitted) on limits shows that X = \mathop{\mathrm{lim}}\nolimits X_ J as topological spaces. \square
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