Definition dense (S:Ensemble (point_set X)) : Prop :=
closure S = Full_set.

(* Specifications: *)
Lemma dense_meets_every_nonempty_open:
forall S:Ensemble (point_set X), dense S ->
forall U:Ensemble (point_set X), open U -> Inhabited U ->
Inhabited (Intersection S U).
Lemma meets_every_nonempty_open_impl_dense:
forall S:Ensemble (point_set X),
(forall U:Ensemble (point_set X), open U -> Inhabited U ->
Inhabited (Intersection S U)) ->
dense S.