The topological types of hyperspaces of 0-dimensional compacta

Abstract

Let X be a 0-dimensional compactum (= compact metric space) and let 2 X denote the hyperspace of X, the set of non-empty closed subsets of X with the Hausdorff metric. We shall determine all the topological types of the hyperspaces 2 X . They are only finitely many in the non-trivial case where the scattered part of X contains infinitely many points.  2004 Published by Elsevier B.V. MSC: 54B20; 54E45Keywords: Hyperspace; 0-dimensional compactum IntroductionLet X be a 0-dimensional compactum (= compact metric space) and let 2 X denote the set of non-empty closed subsets of X with the Hausdorff metric. As well known, 2 X is a 0-dimensional compactum again, and if X is a Cantor set then so is 2 X . By the CantorBendixson theorem, X is uniquely decomposed as X = K ∪ S with K a closed set of X having no isolated points and S a scattered space. The kernel K, if not empty, is homeomorphic to the Cantor set. We also use the symbol D to denote the isolated points of X. Note that S D, with D the closure of D.In the special case where S is dense in X, the structure of 2 X was studied by Pelczyński and found to be uniquely determined if S contains infinitely many points ([3]; also see [2]).

Free Modal Algebras: A Coalgebraic Perspective

“…(see [28] and [26]) A space P is called the Pelczynski space if P = X iso ∪ X lim , where X iso is a countable set of isolated points of P, X lim is the set of limit points of P, the space X lim is homeomorphic to the Cantor space C and X iso = P.…”

“…We also observe that the underlying Stone space of the canonical model of the basic modal logic is homeomorphic to the so-called Pelczynski space. This space appears to be one of the nine fixed points of the Vietoris functor on compact Hausdorff spaces with a countable basis [28,26].…”