Stone Duality for Topological Convexity Spaces

TL;DR

This paper extends Stone duality to topological convexity spaces, establishing an adjunction between these spaces and sup-lattices, thereby linking convexity, topology, and lattice theory.

Contribution

It introduces a duality framework connecting topological convexity spaces with sup-lattices, generalizing existing dualities in topology and lattice theory.

Findings

Extended Stone duality to topological convexity spaces.

Established an adjunction between topological convexity spaces and sup-lattices.

Connected convexity, topology, and lattice structures in a unified framework.

Abstract

A convexity space is a set X with a chosen family of subsets (called convex subsets) that is closed under arbitrary intersections and directed unions. There is a lot of interest in spaces that have both a convexity space and a topological space structure. In this paper, we study the category of topological convexity spaces and extend the Stone duality between coframes and topological spaces to an adjunction between topological convexity spaces and sup-lattices. We factor this adjunction through the category of preconvexity spaces (somtimes called closure spaces).