Lemma is_lim_base_correct {X : UU} {T : TopologicalSpace} (f : X → T) (F : Filter X) (x : T) base : is_lim_base f F x base <-> is_lim f F x. Proof. split. - intros Hx P HP. apply Hx, (pr2 (neighborhood_equiv _ _ _)). exact HP. - intros Hx P HP. eapply Hx, neighborhood_equiv. exact HP. 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 _ _).