Lemma ex_filter_lim_base_correct {T : TopologicalSpace} (F : Filter T) : ex_filter_lim_base F <-> ex_filter_lim F. Proof. split. - apply hinhfun. intros x. exists (pr1 x). eapply is_filter_lim_base_correct. exact (pr2 (pr2 x)). - apply hinhfun. intros x. exists (pr1 x), (base_of_neighborhood_default (pr1 x)). apply (pr2 (is_filter_lim_base_correct _ _ _)). exact (pr2 x). 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 _ _).