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.

Lemma meets_every_open_neighborhood_impl_closure:
forall (S:Ensemble (point_set X)) (x:point_set X),
(forall U:Ensemble (point_set X), open U -> In U x ->
Inhabited (Intersection S U)) ->
In (closure S) x.