Lemma isNeighborhood_neighborhood {T : TopologicalSpace} : isNeighborhood (neighborhood (T := T)). Proof. repeat split. - intros x A B. apply (neighborhood_imply x). - intros x. apply (pr2 (neighborhood_isOpen _)). + exact (isOpen_htrue (T := T)). + apply tt. - intros A B. apply neighborhood_and. - intros x P. apply neighborhood_point. - intros x P. apply neighborhood_neighborhood. 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 _ _).