Lemma isNeighborhood_isPreFilter {X : hSet} N : isNeighborhood N -> ∏ x : X, isPreFilter (N x). Proof. intros Hn x. split. - apply (pr1 Hn). - apply isfilter_finite_intersection_carac. + apply (pr1 (pr2 Hn)). + apply (pr1 (pr2 (pr2 Hn))). 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 _ _).