Countable quasicontinuous domains are quasialgebraic

TL;DR

The paper addresses the gap between quasicontinuous and quasialgebraic domains in dcpos by proving a Urysohn-style result: if a quasicontinuous domain $P$ is not quasialgebraic, there exists a surjective monotone Lawson-continuous map $f: P \to $ $[0,1]$. The construction hinges on an interpolation framework on finite subsets of $P$ and a dyadic subdivision of the unit interval, yielding a function that is monotone and Lawson-continuous. This leads to the key corollary that every countable quasicontinuous domain is quasialgebraic, and, together with additional topology-meet-continuity arguments, recovers that every countable continuous domain is algebraic. The approach blends Rudin's lemma-inspired techniques with domain-theoretic interpolation, enhancing our understanding of how approximation properties determine (quasi)algebraicity in countable domains and providing a constructive path between these classes.

Abstract

We prove that every quasicontinuous domain that fails to be quasialgebraic admits the unit interval [0, 1] as its monotone Lawson-continuous image. As a result, every countable quasicontinuous domain is quasialgebraic.

Theorems & Definitions (19)

• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8
• Definition 2.9
• Lemma 2.10