The reflective hull of the two-element chain in DCPO: properness, maximal Γ-faithfulness, and an internal reflection formula

Let $\text{DCPO}$ be the category of dcpos and Scott-continuous maps, and let $\text{Rtwo}$ be the reflective hull of the two-element chain $2$. We prove that $\text{Rtwo}$ is the least non-discrete proper reflective full subcategory of $\text{DCPO}$. It is closed under limits and Skula-closed sub-dcpos and contains every sober dcpo. We show that $\text{Rtwo}$ is the maximal $\mathrm{\Gamma }$-faithful full subcategory of $\text{DCPO}$ that strictly contains weakly dominated dcpos. Finally, we give the concrete construction of $2$-reflection internally by a transfinite iteration and a criterion for reflective full subcategories of $\text{DCPO}$, analogous to the Keimel–Lawson conditions for $T_0$ topological spaces.