Lemma neighborhood_imply : ∏ (x : T) (P Q : T → hProp), (∏ y : T, P y → Q y) → neighborhood x P → neighborhood x Q. Proof. intros x P Q H. apply hinhfun. intros O. exists (pr1 O). split. - apply (pr1 (pr2 O)). - intros y Hy. apply H. apply (pr2 (pr2 O)). exact Hy. 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 _ _).