proposition: Lemma TopologyGenerated_smallest {X : hSet} : ∏ (O : (X → hProp) → hProp) (T : isTopologicalSpace X), (∏ P : X → hProp, O P → pr1 T P) → ∏ P : X → hProp, isOpen (T := TopologyGenerated O) P → pr1 T P.
proof: Proof. intros O T Ht P Hp. apply (neighborhood_isOpen (T := (X,,T))). intros x Px. generalize (Hp x Px) ; clear Hp. apply hinhfun. intros L. simple refine (tpair _ _ _). - simple refine (tpair _ _ _). + apply (finite_intersection (pr1 L)). + apply (isOpen_finite_intersection (T := X,,T)). intros m. apply Ht. apply (pr1 (pr2 L)). - split. + exact (pr1 (pr2 (pr2 L))). + exact (pr2 (pr2 (pr2 L))). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.Topology.Filters UniMath.Algebra.Groups UniMath.Algebra.DivisionRig UniMath.Algebra.ConstructiveStructures