The Stacks project

Lemma 33.8.16. Let $X$ be a scheme over the field $k$. Assume $X$ has finitely many irreducible components which are all geometrically irreducible. Then $X$ has finitely many connected components each of which is geometrically connected.

Proof. This is clear because a connected component is a union of irreducible components. Details omitted. $\square$

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