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On Kripke, Vietoris and Hausdorff Polynomial Functors ((Co)algebraic pearls) Adámek, Jiří Milius, Stefan Moss, Lawrence S. Hausdorff functor Vietoris functor initial algebra terminal coalgebra The Vietoris space of compact subsets of a given Hausdorff space yields an endofunctor V on the category of Hausdorff spaces. Vietoris polynomial endofunctors on that category are built from V, the identity and constant functors by forming products, coproducts and compositions. These functors are known to have terminal coalgebras and we deduce that they also have initial algebras. We present an analogous class of endofunctors on the category of extended metric spaces, using in lieu of V the Hausdorff functor ℋ. We prove that the ensuing Hausdorff polynomial functors have terminal coalgebras and initial algebras. Whereas the canonical constructions of terminal coalgebras for Vietoris polynomial functors takes ω steps, one needs ω + ω steps in general for Hausdorff ones. We also give a new proof that the closed set functor on metric spaces has no fixed points. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jiří Adámek and Stefan Milius and Lawrence S. Moss 2023 Is Part Of LIPIcs, Volume 270, 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CALCO.2023.21 urn:nbn:de:0030-drops-188189 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.21 eng https://creativecommons.org/licenses/by/4.0/legalcode