Definition to_hausdorff_separated_continuous_dcpo {X : dcpo} (B : dcpo_basis X) (x y : X) : (∃ (b₁ b₂ : B), B b₁ ≪ x ∧ B b₂ ≪ y ∧ ¬(∃ (t : B), B b₁ ≪ B t ∧ B b₂ ≪ B t))%logic → is_hausdorff_separated x y.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets. Require Import UniMath.OrderTheory.DCPOs.Core.Basics. Require Import UniMath.OrderTheory.DCPOs.Core.WayBelow. Require Import UniMath.OrderTheory.DCPOs.Core.ScottTopology. Require Import UniMath.OrderTheory.DCPOs.Core.IntrinsicApartness. Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous. Require Import UniMath.OrderTheory.DCPOs.Basis.Basis. Require Import UniMath.OrderTheory.DCPOs.Elements.Sharp.