Lemma TopologyGenerated_included {X : hSet} : ∏ (O : (X → hProp) → hProp) (P : X → hProp), O P → isOpen (T := TopologyGenerated O) P. Proof. intros O P Op. apply neighborhood_isOpen. intros x Hx. apply TopologyFromNeighborhood_correct. apply hinhpr. exists (singletonSequence P). repeat split. - induction n as [n Hn]. exact Op. - intros n ; induction n as [n Hn]. exact Hx. - intros y Hy. now apply (Hy (0%nat,,paths_refl _)). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).