Definition Build_TopologicalSpace_from_closure_operator : TopologicalSpace.
refine (Build_TopologicalSpace_from_closed_sets
(fun F => cl F = F) _ _ _).
apply cl_empty.
intros.
rewrite cl_respects_union; congruence.
intros.
apply Extensionality_Ensembles; split; try apply cl_inflationary.
red; intros.
constructor; intros.
rewrite <- (H S H1).
apply cl_increasing with (FamilyIntersection F); trivial.
red; intros.
destruct H2.
apply H2; trivial.
Defined.

Lemma Build_TopologicalSpace_from_closure_operator_closure:
forall S:Ensemble (point_set Build_TopologicalSpace_from_closure_operator),
closure S = cl S.