Section interior_closure_relations.
Definition idempotent {T:Type} (f:T->T) := forall x, f (f x) = f x.
Hint Unfold idempotent : closure.
Variable X:TopologicalSpace.
Definition dense (S:Ensemble (point_set X)) : Prop :=
closure S = Full_set.

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.