Definition locally {T : TopologicalSpace} (x : T) : Filter T. Proof. simple refine (make_Filter _ _ _ _ _). - apply (neighborhood x). - abstract (intros A B ; apply neighborhood_imply). - abstract (apply (pr2 (neighborhood_isOpen _)) ; [ apply isOpen_htrue | apply tt]). - abstract (intros A B ; apply neighborhood_and). - abstract (intros A Ha ; apply hinhpr ; exists x ; now apply neighborhood_point in Ha). 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 _ _).