Lemma topologyfromneighborhood_open : isSetOfOpen_union (λ A : X → hProp, make_hProp (topologyfromneighborhood A) (isaprop_topologyfromneighborhood A)). Proof. intros L Hl x. apply hinhuniv. intros A. apply Himpl with (pr1 A). - intros y Hy. apply hinhpr. now exists (pr1 A), (pr1 (pr2 A)). - apply Hl. + exact (pr1 (pr2 A)). + exact (pr2 (pr2 A)). 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 _ _).