Constructible subgroup is closed
= Constructible subgroup is closed
Let $S$ be constructible in an <algebraic group>, and $C=\overline S$. The closure is a subgroup. Choose a dense open $U\subseteq C$ contained in $S$. For every $c\in C$, the dense opens $U,cU$ intersect, giving $u=cv$ for $u,v\in S$. Hence $c=uv^{-1}\in S$ and $S=C$. The <Chevalley constructibility theorem> therefore proves closedness of homomorphism images.