End interior_closure.
Arguments interior {X}.
Arguments closure {X}.
Arguments interior_open {X}.
Arguments interior_deflationary {X}.
Arguments interior_fixes_open {X}.
Arguments interior_maximal {X}.
Arguments closure_closed {X}.
Arguments closure_inflationary {X}.
Arguments closure_fixes_closed {X}.
Arguments closure_minimal {X}.
Section interior_closure_relations.
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.