Definition locally_base {T : TopologicalSpace} (x : T) (base : base_of_neighborhood x) : Filter T. Proof. simple refine (make_Filter _ _ _ _ _). - apply (neighborhood' x base). - apply locally_base_imply. - apply locally_base_htrue. - apply locally_base_and. - intros A Ha. apply neighborhood_equiv in Ha. apply neighborhood_point in Ha. apply hinhpr. now exists x. Defined.
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 _ _).