Section interior_closure.
Variable X:TopologicalSpace.
Variable S:Ensemble (point_set X).
Definition interior := FamilyUnion
[ U:Ensemble (point_set X) | open U /\ Included U S ].
Definition closure := FamilyIntersection
[ F:Ensemble (point_set X) | closed F /\ Included S F ].

Lemma interior_maximal: forall U:Ensemble (point_set X),
open U -> Included U S -> Included U interior.