Lemma 10.48.5. Let $k$ be a field. Let $S$ be a $k$-algebra.
If $S$ is geometrically connected over $k$ so is every $k$-subalgebra.
If all finitely generated $k$-subalgebras of $S$ are geometrically connected, then $S$ is geometrically connected.
A directed colimit of geometrically connected $k$-algebras is geometrically connected.