Lemma base_default_2 : ∏ P : T → hProp, neighborhood x P → ∃ Q : T → hProp, base_default Q × (∏ t : T, Q t → P t). Proof. intros P. apply hinhfun. intros Q. exists (pr1 Q). repeat split. - exact (pr2 (pr1 Q)). - exact (pr1 (pr2 Q)). - exact (pr2 (pr2 Q)). 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 _ _).