Lemma isNeighborhood_isFilter {X : hSet} N : isNeighborhood N -> ∏ x : X, isFilter (N x). Proof. intros Hn x. split. - apply isNeighborhood_isPreFilter, Hn. - intros A Fa. apply hinhpr. exists x. apply ((pr1 (pr2 (pr2 (pr2 Hn)))) _ _ Fa). 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 _ _).