Lemma continuous_tpair {U V : TopologicalSpace} : continuous2d (W := TopologyDirprod U V) (λ (x : U) (y : V), (x,,y)). Proof. intros x y P. apply hinhuniv. intros O. simple refine (filter_imply _ _ _ _ _). - exact (pr1 O). - exact (pr2 (pr2 O)). - generalize (pr2 (pr1 O) _ (pr1 (pr2 O))). apply hinhfun. intros Ho. exists (pr1 Ho), (pr1 (pr2 Ho)). repeat split. + apply (pr2 (neighborhood_isOpen _)). * exact (pr2 (pr1 (pr2 (pr2 Ho)))). * exact (pr1 (pr1 (pr2 (pr2 Ho)))). + apply (pr2 (neighborhood_isOpen _)). * exact (pr2 (pr1 (pr2 (pr2 (pr2 Ho))))). * exact (pr1 (pr1 (pr2 (pr2 (pr2 Ho))))). + exact (pr2 (pr2 (pr2 (pr2 Ho)))). 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 _ _).