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We generalize the classic Vietoris endofunctor to the category of compactHausdorff spaces and closed relations. The lift of a closed relation is done bygeneralizing the construction of the Egli-Milner order. We describe the dualendofunctor on the category of de Vries algebras and subordinations. This isdone in several steps, by first generalizing the construction of Venema andVosmaer to the category of boolean algebras and subordinations, then lifting itup to $\mathsf{S5}$-subordination algebras, and finally using MacNeillecompletions to further lift it to de Vries algebras. Among other things, thisyields a generalization of Johnstone's pointfree construction of the Vietorisendofunctor to the category of compact regular frames and preframehomomorphisms.

Vietoris endofunctor for closed relations and its de Vries dual