Directed Convex Powerspaces and Convex Powerdomains

Abstract

It is known that lower powerdomains preserve and reflect both continuity and quasicontinuity, while the preservation of quasicontinuity by upper and convex powerdomains had long been open. Directed spaces provide a topological extension framework for dcpos. Powerdomains of dcpos can be characterized as the $D$-completions of the corresponding directed powerspaces. Using this observation, the authors proved in 2024 that upper powerdomains do not preserve quasicontinuity. In this paper, we prove that directed convex powerspaces and convex powerdomains do not preserve quasicontinuity, but they do reflect both continuity and quasicontinuity. We also prove that upper powerspaces and upper powerdomains reflect quasicontinuity. Together with the known results for lower and upper powerdomains, these arguments give the complete preservation and reflection profile of the lower, upper and convex powerdomains for continuity and quasicontinuity.