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]).
Abstract. We define a subclass, denoted by EM3, of the class of stratifiable spaces, and obtain several dimension theoretical results for EM3 including the coincidence theorem for dim and Ind. The class EM3 is countably productive, hereditary, preserved under closed maps and, furthermore, the largest subclass of stratifiable spaces in which a harmonious dimension theory can be established.
Abstract. We define a subclass, denoted by EM3, of the class of stratifiable spaces, and obtain several dimension theoretical results for EM3 including the coincidence theorem for dim and Ind. The class EM3 is countably productive, hereditary, preserved under closed maps and, furthermore, the largest subclass of stratifiable spaces in which a harmonious dimension theory can be established.
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