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Stone Duality for Preordered Topological Spaces

Jean Goubault-Larrecq

TL;DR

This work develops a Stone-like duality for preordered topological spaces by introducing ad-frames, a bipartite pointfree structure linking frames with completely distributive lattices through tot, con, sup, and sub relations. It defines the adjunction $\mathcal{O}^{ad} \dashv \textbf{pt}^{ad}$ between $\mathbf{PreTop}$ and $\mathbf{adFrm}^{op}$, and constructs ad-sobrification $X^{ads}$ with unit $\eta_X$, showing $\textbf{pt}^{ad}\, \mathcal{O}^{ad} X \cong X^{ads}$; this route generalizes classical Stone duality while avoiding point-based conditions on the dual side. The theory also relates soberification to ad-soberification, proving the idempotence of the adjunction via a monad $\_^{ads}$ and clarifying how the red/purple/blue variants capture future/past or both viewpoints. Moreover, the adjunction lifts the standard $\mathcal{O} \dashv \textbf{pt}$ adjunction, yielding a topological-functorial comparison and highlighting cases where the lifting aligns with sobriety. Overall, the framework opens avenues toward ad-locales and constructive logic applications, and points toward extensions to streams and ordered-logical semantics.

Abstract

A preordered topological space is a topological space with a preordering. We exhibit a Stone-like duality for preordered topological spaces, Inspired by a similar duality for bitopological spaces, due to Jung-Moshier and Jakl, and by a duality for preordered sets due to Bonsangue, Jacobs and Kok.

Stone Duality for Preordered Topological Spaces

TL;DR

This work develops a Stone-like duality for preordered topological spaces by introducing ad-frames, a bipartite pointfree structure linking frames with completely distributive lattices through tot, con, sup, and sub relations. It defines the adjunction between and , and constructs ad-sobrification with unit , showing ; this route generalizes classical Stone duality while avoiding point-based conditions on the dual side. The theory also relates soberification to ad-soberification, proving the idempotence of the adjunction via a monad and clarifying how the red/purple/blue variants capture future/past or both viewpoints. Moreover, the adjunction lifts the standard adjunction, yielding a topological-functorial comparison and highlighting cases where the lifting aligns with sobriety. Overall, the framework opens avenues toward ad-locales and constructive logic applications, and points toward extensions to streams and ordered-logical semantics.

Abstract

A preordered topological space is a topological space with a preordering. We exhibit a Stone-like duality for preordered topological spaces, Inspired by a similar duality for bitopological spaces, due to Jung-Moshier and Jakl, and by a duality for preordered sets due to Bonsangue, Jacobs and Kok.
Paper Structure (10 sections, 17 theorems)

This paper contains 10 sections, 17 theorems.

Table of Contents

  1. Introduction
  2. Basic idea, preliminaries, and outline
  3. Ad-frames
  4. The $\mathcal{O}^{ad} \dashv \textbf{pt}^{ad}\,$ adjunction
  5. Ad-sobrification
  6. Sobrification and ad-sobrification
  7. The $\mathcal{O}^{ad} \dashv \textbf{pt}^{ad}\,$ adjunction is idempotent
  8. The $\mathcal{O}^{ad} \dashv \textbf{pt}^{ad}\,$ adjunction lifts the $\mathcal{O} \dashv \textbf{pt}\,$ adjunction
  9. Conclusion
  10. Acknowledgments

Key Result

Lemma 3.4

For every continuous, order-preserving map $f \colon X \to Y$ between preordered topological spaces, the pair $\mathcal{O}^{ad} f \mathrel{\buildrel \text{def}\over=} (f^{-1}, f^{-1})$ is an ad-frame homomorphism from $\mathcal{O}^{ad} Y$ to $\mathcal{O}^{ad} X$. Here $f^{-1}$ denotes the function t

Theorems & Definitions (49)

  • Definition 3.1
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • Definition 4.1
  • Remark 4.2
  • Remark 4.3
  • Lemma 4.4
  • ...and 39 more