Vietoris endofunctor for closed relations and its de Vries dual

TL;DR

This paper extends the Vietoris endofunctor to compact Hausdorff spaces and closed relations, providing a dual construction on de Vries algebras and generalizing pointfree topology methods.

Contribution

It introduces a generalized Vietoris endofunctor for closed relations and develops its dual on de Vries algebras, expanding the categorical and algebraic framework.

Findings

Generalizes Vietoris endofunctor to compact Hausdorff spaces

Constructs dual endofunctor on de Vries algebras and subordinations

Extends pointfree topology to regular frames

Abstract

We generalize the classic Vietoris endofunctor to the category of compact Hausdorff spaces and closed relations. The lift of a closed relation is done by generalizing the construction of the Egli-Milner order. We describe the dual endofunctor on the category of de Vries algebras and subordinations. This is done in several steps, by first generalizing the construction of Venema and Vosmaer to the category of boolean algebras and subordinations, then lifting it up to $\mathsf{S5}$-subordination algebras, and finally using MacNeille completions to further lift it to de Vries algebras. Among other things, this yields a generalization of Johnstone's pointfree construction of the Vietoris endofunctor to the category of compact regular frames and preframe homomorphisms.