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 has finite index in some
parabolic subgroup. 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.