The Stacks project

Lemma 28.5.7. A Noetherian scheme has a finite number of irreducible components.

Proof. The underlying topological space of a Noetherian scheme is Noetherian (Lemma 28.5.5) and we conclude because a Noetherian topological space has only finitely many irreducible components (Topology, Lemma 5.9.2). $\square$

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