Notes on countable frames

TL;DR

This paper investigates the conditions under which countable frames are continuous lattices, establishing links between continuity, quasicontinuity, and spectral properties, and addressing an open problem in domain theory.

Contribution

It proves that continuity of countable frames implies quasicontinuity of spectra and identifies conditions for spectra to be Scott spaces, partially resolving an open problem.

Findings

Continuity implies quasicontinuity of spectra in countable frames

Spectra are Scott spaces under certain conditions such as T1 or Scott spaces

Existence of non-continuous countable frames confirmed

Abstract

Matthew de Brecht raised the question of whether countable frames are continuous lattices. We prove that the continuity of a countable frame implies the quasicontinuity of its corresponding spectrum in the dual specialization order. We further show that this question admits a positive answer if the frame's spectrum is a $T_1$ space or a Scott space. In general, we confirm the existence of non-continuous countable frames. This work also partially addresses an open problem proposed by Jimmie Lawson and Michael Mislove in 1990, which concerns the characterization of when the spectrums of spatial frames are Scott spaces.