Priestley perspective on pointfree topology

TL;DR

This survey develops a unified Priestley-space perspective on pointfree topology by embedding frames into the world of $L$-spaces and exploiting kernels, nuclei, and sublocales. It extends Priestley duality to frames, analyzes spatiality via dense localic parts, and provides uniform kernel-based treatments of subfitness, Hausdorffness, regularity, and normality. It proves Isbell duality for compact regular frames and Hofmann–Lawson duality for continuous frames, linking these algebras to locally compact and locally compact Hausdorff spaces, respectively. Finally, it derives the classical Priestley and Stone dualities from the algebraic, coherent, and Stone-frame perspectives, illustrating how pointfree topology can be understood through spectral Priestley spaces and their localic-topological interplay.

Abstract

Priestley duality has diverse applications in various branches of mathematics. In this survey, we discuss its usefulness in pointfree topology. This is done by providing Priestley perspective on several key notions, including spatiality, sublocales, separation axioms, compactness, and local compactness. This approach yields a new perspective on a number of classic results in pointfree topology.

Figures (3)

• Figure 1: Equivalences and adjunctions between frames, $\mathsf{L}$-spaces, and topological spaces, together with their spatial/sober restrictions.
• Figure 2: Equivalences between continuous frames, continuous $\mathsf{L}$-spaces, and locally compact sober spaces, together with their restrictions.
• Figure 3: Recovering Priestley and Stone dualities from the dualities for coherent and Stone frames.

Theorems & Definitions (175)

• Definition 2.1: Priestley spaces
• Theorem 2.2: Priestley duality
• Remark 2.3
• Lemma 2.4: Basic facts about Priestley spaces
• Remark 2.5
• Lemma 2.6: Pri70; see also Pri72
• Proposition 2.7: Pri72
• Remark 2.8
• Definition 2.9
• Remark 2.10