Lemma topologysubtype_and : ∏ x : ∑ x : T, dom x, isfilter_and (topologysubtype x). Proof. intros x A B. apply hinhfun2. intros A' B'. exists (λ x, pr1 A' x ∧ pr1 B' x). repeat split. - exact (pr1 (pr1 (pr2 A'))). - exact (pr1 (pr1 (pr2 B'))). - apply isOpen_and. + exact (pr2 (pr1 (pr2 A'))). + exact (pr2 (pr1 (pr2 B'))). - apply (pr2 (pr2 A')), (pr1 X). - apply (pr2 (pr2 B')), (pr2 X). 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 _ _).