De Vries duality for compactifications and completely regular spaces

TL;DR

This paper extends de Vries duality from compact Hausdorff spaces to completely regular spaces by developing a duality framework involving de Vries extensions and compactifications.

Contribution

It introduces a new duality for completely regular spaces using extensions of de Vries algebras, broadening the scope of the original duality.

Findings

Established a duality between compactifications and de Vries extensions.

Specialized the duality to Stone-ech compactifications.

Extended the de Vries duality to a larger class of spaces.

Abstract

De Vries duality yields a dual equivalence between the category of compact Hausdorff spaces and a category of complete Boolean algebras with a proximity relation on them, known as de Vries algebras. We extend de Vries duality to completely regular spaces by replacing the category of de Vries algebras with certain extensions of de Vries algebras. This is done by first formulating a duality between compactifications and de Vries extensions, and then specializing to the extensions that correspond to Stone-\v{C}ech compactifications.