Section Build_from_interior.
Variable X:Type.
Variable int : Ensemble X -> Ensemble X.
Hypothesis int_deflationary: forall S:Ensemble X,
Included (int S) S.
Hypothesis int_respects_intersection: forall S1 S2:Ensemble X,
int (Intersection S1 S2) = Intersection (int S1) (int S2).
Hypothesis int_full: int Full_set = Full_set.
Hypothesis int_idempotent: forall S:Ensemble X,
int (int S) = int S.
Definition Build_TopologicalSpace_from_interior_operator : TopologicalSpace.
apply Build_TopologicalSpace with (point_set:=X) (open:=fun F => int F = F);
intros.
- extensionality_ensembles.
+ now apply int_deflationary.
+ eapply int_family_union; [ eassumption | ].
now rewrite H.
- rewrite <- H, <- H0 at 2.
apply int_respects_intersection.
- assumption.
Defined.

Lemma Build_TopologicalSpace_from_interior_operator_interior:
forall S:Ensemble (point_set Build_TopologicalSpace_from_interior_operator),
interior S = int S.