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.
Lemma cl_increasing: forall S1 S2:Ensemble X, Included S1 S2 -> Included (cl S1) (cl S2).