End interior_closure_relations.
Arguments interior_increasing {X}.
Arguments interior_intersection {X}.
Arguments interior_union {X}.
Arguments interior_complement {X}.
Arguments closure_increasing {X}.
Arguments closure_complement {X}.
Arguments closure_union {X}.
Arguments closure_intersection {X}.
Arguments dense {X}.
Section Build_from_closure.
Variable X:Type.
Variable cl : Ensemble X -> Ensemble X.
Hypothesis cl_inflationary: forall S:Ensemble X,
Included S (cl S).
Hypothesis cl_respects_union: forall S1 S2:Ensemble X,
cl (Union S1 S2) = Union (cl S1) (cl S2).
Hypothesis cl_empty: cl Empty_set = Empty_set.
Hypothesis cl_idempotent: forall S:Ensemble X,
cl (cl S) = cl S.

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