Apartness, sharp elements, and the Scott topology of domains

Abstract Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness sharp elements. Being apart is positive formulation being unequal, similar to how inhabitedness nonemptiness. To exemplify sharpness, note that lower real if only it located. first main result for large class dcpos, Bridges–Vîţǎ topology coincide. Although cannot expect tight or cotransitive on nontrivial prove both when restricted elements dcpo. These include strongly maximal elements, as studied by Smyth Heckmann. We develop theory highlighting its connection sharpness Lawson Finally, illustrate apartness, strong maximality considering several natural examples dcpos: Cantor Baire domains, partial Dedekind reals, reals and, finally, an embedding space into exponential lifted sets.