Lemma topologygenerated_imply : ∏ x : X, isfilter_imply (topologygenerated x). Proof. intros x A B H. apply hinhfun. intros L. exists (pr1 L). repeat split. - exact (pr1 (pr2 L)). - exact (pr1 (pr2 (pr2 L))). - intros y Hy. apply H, (pr2 (pr2 (pr2 L))), Hy. 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 _ _).