Definition is_base_of_neighborhood {T : TopologicalSpace} (x : T) (B : (T → hProp) → hProp) := (∏ P : T → hProp, B P → neighborhood x P) × (∏ P : T → hProp, neighborhood x P → ∃ Q : T → hProp, B Q × (∏ t : T, Q t → P t)).
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 _ _).