Lemma 27.5.5. A (locally) Noetherian scheme has a (locally) Noetherian underlying topological space, see Topology, Definition 5.9.1.

**Proof.**
This is because a Noetherian scheme is a finite union of spectra of Noetherian rings and Algebra, Lemma 10.30.5 and Topology, Lemma 5.9.4.
$\square$

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