Locally Compact Stone Duality

TL;DR

This paper establishes dualities between posets and bases of open sets in locally compact Hausdorff spaces, characterizing various substructures and their properties through axiomatization and representation theory.

Contribution

It introduces new dualities and characterizations for sublattices and pseudobasic posets in locally compact spaces, expanding the theoretical framework of Stone duality.

Findings

Relatively compact basic sublattices are finitely axiomatizable.

Relatively compact basic subsemilattices omit certain types.

Compact clopen pseudobasic posets are characterized by separativity.

Abstract

We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic subsemilattices are those omitting certain types. (3) Compact clopen pseudobasic posets are characterized by separativity. We also show how to obtain the tight spectrum of a poset as the Stone space of a generalized Boolean algebra that is universal for tight representations.