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Notes on countable frames

Xiaodong Jia, Xiaoyong Xi

TL;DR

The paper investigates whether countable frames must be continuous lattices by examining the spectrum in the dual specialization order. It proves that frame continuity implies spectrum quasicontinuity, and that a positive answer holds when the spectrum is $T_1$ or a Scott space; counterexamples then show that countable frames can be non-continuous in general. It also provides a partial resolution to Lawson–Mislove by showing that for frames that arise as Scott-open lattices of dcpos, the frame is continuous and the spectrum is a Scott space, with several equivalent conditions tying frame properties to its spectrum. Together, these results map out when a countable frame’s spectrum can be a Scott space and clarify the scope of the Lawson–Mislove question beyond special cases.

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.

Notes on countable frames

TL;DR

The paper investigates whether countable frames must be continuous lattices by examining the spectrum in the dual specialization order. It proves that frame continuity implies spectrum quasicontinuity, and that a positive answer holds when the spectrum is or a Scott space; counterexamples then show that countable frames can be non-continuous in general. It also provides a partial resolution to Lawson–Mislove by showing that for frames that arise as Scott-open lattices of dcpos, the frame is continuous and the spectrum is a Scott space, with several equivalent conditions tying frame properties to its spectrum. Together, these results map out when a countable frame’s spectrum can be a Scott space and clarify the scope of the Lawson–Mislove question beyond special cases.

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 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.
Paper Structure (8 sections, 8 theorems, 1 equation, 1 figure)

This paper contains 8 sections, 8 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Order and topology
  4. Domain theory
  5. Stone duality
  6. Lawson and Mislove's open question
  7. The general case
  8. A partial answer to Lawson and Mislove's question

Key Result

Lemma 3.1

Let $X$ be a sober topological space. If $C$ is a closed subset of $X$ and $\max C$ is infinite, then the open set lattice $\mathcal{O}(X)$ of $X$ is uncountable.

Figures (1)

  • Figure 1: $P$

Theorems & Definitions (17)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 7 more