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 _)).

(* Specifications: *)
Lemma expand_U_dyadic_open: forall f Hopen Hincr n,
open (expand_U_dyadic f Hopen Hincr n).
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)).