The Stacks project

Lemma 29.16.7. Let $S$ be a scheme. For any locally closed subset $T \subset S$ we have

\[ T \not= \emptyset \Rightarrow T \cap S_{\text{ft-pts}} \not= \emptyset . \]

In particular, for any closed subset $T \subset S$ we see that $T \cap S_{\text{ft-pts}}$ is dense in $T$.

Proof. Note that $T$ carries a scheme structure (see Schemes, Lemma 26.12.4) such that $T \to S$ is a locally closed immersion. Any locally closed immersion is locally of finite type, see Lemma 29.15.5. Hence by Lemma 29.16.5 we see $T_{\text{ft-pts}} \subset S_{\text{ft-pts}}$. Finally, any nonempty affine open of $T$ has at least one closed point which is a finite type point of $T$ by Lemma 29.16.4. $\square$

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