The Stacks project

Lemma 31.26.1. Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. The collection of irreducible components of $Z$ is locally finite in $X$.

Proof. Let $U \subset X$ be a quasi-compact open subscheme. Then $U$ is a Noetherian scheme, and hence has a Noetherian underlying topological space (Properties, Lemma 28.5.5). Hence every subspace is Noetherian and has finitely many irreducible components (see Topology, Lemma 5.9.2). $\square$

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