Definition continuous_on {U V : TopologicalSpace} (dom : U → hProp) (f : ∏ (x : U), dom x → V) := ∏ (x : U) (Hx : dom x), ∃ H, is_lim (λ y : (∑ x : U, dom x), f (pr1 y) (pr2 y)) (FilterSubtype (locally x) dom H) (f x Hx).
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 _ _).