Definition TopologySubtype {T : TopologicalSpace} (dom : T → hProp) : TopologicalSpace. Proof. simple refine (TopologyFromNeighborhood _ _). - exact (carrier_subset dom). - apply topologysubtype. - repeat split. + apply topologysubtype_imply. + apply topologysubtype_htrue. + apply topologysubtype_and. + apply topologysubtype_point. + apply topologysubtype_neighborhood. Defined.
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 _ _).