[Paper Review] Stone Duality for Preordered Topological Spaces

Jean Goubault-Larrecq|arXiv (Cornell University)|Jan 19, 2026

Fuzzy and Soft Set Theory0 citations

TL;DR

The paper develops a Stone-like duality for preordered topological spaces by introducing ad-frames and an adjunction between PreTop and adFrm, extending classical Stone duality to ordered settings without restricting to compactness.

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.

Motivation & Objective

• Motivate and formalize a duality for preordered topological spaces extending Stone duality beyond compact or pointed cases.
• Fuse ideas from Alexandroff spaces, bitopological dualities, and ordered locales into a unified adjunction framework.
• Provide a concrete, point-free (locale-like) approach that avoids point-based constraints on dual categories.

Proposed method

• Define ad-frames as a quadruple with two frames and two relations capturing totality and consistency and their interactions.
• Construct the ad-Frame category adFrm and demonstrate it as the target of a duality adjunction from PreTop via O^{ad} and pt^{ad}.
• Show that O^{ad} extends the usual O functor to an adjunction with pt^{ad}, forming O^{ad} dashv pt^{ad}.
• Provide canonical instances and verify that O^{ad}X forms a valid ad-frame for any preordered space X.
• Establish that every ad-frame yields a preordered topological space via pt^{ad}, and that there is a corresponding functorial construction in the opposite direction.

Experimental results

Research questions

• RQ1 How can preordered topological spaces be dualized without restricting to compactness or specific subcategories?
• RQ2 What is the appropriate enrichment of frames/locales to capture both topology and order simultaneously?
• RQ3 Can we formulate and prove an adjunction between PreTop and a suitably enriched frame-like structure (ad-frames) that generalizes Stone duality?
• RQ4 How do notions of openness, order, and interaction laws (tot, con, sup, sub) translate into morphisms and points in the dual category?

Key findings

• Introduces ad-frames as a structured pairing of a frame and a completely distributive lattice with interaction relations tot, con, sup, sub.
• Defines the adjunction O^{ad} dashv pt^{ad} between PreTop and adFrm, mirroring Stone duality in a preordered setting.
• Shows that the canonical construction O^{ad}X for a preordered space X yields a valid ad-frame, and that pt^{ad} of an ad-frame is a preordered topological space.
• Proves idempotence of the O^{ad} dashv pt^{ad} adjunction, aligning with traditional dualities and allowing a sobrification-like construction (ad-sobrification).
• Demonstrates that the duality lifts a prior adjunction between ordinary topological spaces and locales, now enriched to handle order without point-based constraints.