The Stacks project

Lemma 72.6.1. Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space over $S$. If $T \subset |X|$ is a closed subset, then the collection of irreducible components of $T$ is locally finite.

Proof. The topological space $|X|$ is locally Noetherian (Properties of Spaces, Lemma 66.24.2). A Noetherian topological space has a finite number of irreducible components and a subspace of a Noetherian space is Noetherian (Topology, Lemma 5.9.2). Thus the lemma follows from the definition of locally finite (Topology, Definition 5.28.4). $\square$

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