Open subgroups of $Gl_n$ which are not closed.

Question

I was reading this post where the OP asks if there are open subgroups of $Gl_n$ that are not closed. The answer uses Lie algebras but I don't understand why. Isn't any open subgroup of a topological group closed and like explained in this question. It seems there is something I am missing? Why take the trouble consider Lie-algebras?

Answer 1

Every open subgroup of every topological group is closed. This is because its complement is the union of its nontrivial coset; each coset is open (as the image of the subgroup under a shift, which is a homeomorphism), so the union of all the nontrivial coset is open, so its complement (i.e. your subgroup) is closed.

You don't need to use Lie algebras or even assume your group is a Lie group.