  Let G be a complete Kac-Moody group over a finite field. It is known that G
possesses a BN-pair structure, all of whose parabolic subgroups are open in G.
We show that, conversely, every open subgroup of G is contained with finite
index in some parabolic subgroup; moreover there are only finitely many such
parabolic subgroups. The proof uses some new results on parabolic closures in
Coxeter groups. In particular, we give conditions ensuring that the parabolic
closure of the product of two elements in a Coxeter group contains the
respective parabolic closures of those elements.
