Strongly Continuous Domains

TL;DR

This paper introduces the concept of strongly continuous domains, exploring their properties, relations to other domain types, and the characteristics of associated topologies, thereby extending the theory of domain topologies.

Contribution

It defines strongly continuous domains, introduces the strong way-below relation, and characterizes these domains, connecting them with continuous and hypercontinuous domains.

Findings

Strong way-below relation satisfies the interpolation property.

Strongly continuous domains are related to both continuous and hypercontinuous domains.

Properties of strong Scott and strong Lawson topologies are analyzed.

Abstract

Strong Scott topology introduced by X. Xu and D. Zhao is a kind of new topology which is finer than upper topology and coarser than Scott topology. Inspired by the topological characterizations of continuous domains and hypercontinuous domains, we introduce the concept of strongly continuous domains and investigate some properties of strongly continuous domains. In particular, we give the definition of strong way-below relation and obtain a characterization of strongly continuous domains via the strong way-below relation. We prove that the strong way-below relation on a strongly continuous domain satisfies the interpolation property, and clarify the relationship between strongly continuous domains and continuous domains, and the relationship between strongly continuous domains and hypercontinuous domains. We discuss the properties of strong Scott topology and strong Lawson topology, which is the common refinement of the strong Scott topology and the lower topology, on a strongly continuous domain.