Sharp Elements and Apartness in Domains

TL;DR

This paper introduces the concepts of intrinsic apartness and sharp elements in continuous dcpos, showing their properties and relationships with existing topologies and maximal elements in domain theory.

Contribution

It defines and analyzes intrinsic apartness and sharpness, demonstrating their equivalence with certain topologies and their behavior on sharp elements in continuous dcpos.

Findings

Intrinsic apartness coincides with the Scott topology on a large class of dcpos.

Intrinsic apartness is tight and cotransitive on sharp elements.

Sharp elements include strongly maximal elements and relate to Lawson topology.

Abstract

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