Definition expand_U_dyadic (f:nat->Ensemble (point_set X))
  (Hopen: forall n:nat, open (f n))
  (Hincr: forall n:nat, Included (closure (f n)) (f (S n)))
  (n:nat) : Ensemble (point_set X) :=
if (even_odd_dec n) then f (div2 n) else
let m := div2 n in proj1_sig
  (normal_sep_fun (closure (f m)) (f (S m))
     (closure_closed _) (Hopen _) (Hincr _)).

Lemma expand_U_dyadic_incr: forall f Hopen Hincr n, Included (closure (expand_U_dyadic f Hopen Hincr n)) (expand_U_dyadic f Hopen Hincr (S n)).