Vietoris endofunctor for closed relations and its de Vries dual

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.