Lemma 29.15.8. Let $S$ be a scheme. The following are equivalent:

the scheme $S$ is Jacobson,

$S_{\text{ft-pts}}$ is the set of closed points of $S$,

for all $T \to S$ locally of finite type closed points map to closed points, and

for all $T \to S$ locally of finite type closed points $t \in T$ map to closed points $s \in S$ with $\kappa (s) \subset \kappa (t)$ finite.

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