On the symmetric hyperspace of the circle |
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Authors: | Naotsugu Chinen |
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Affiliation: | a Hiroshima Institute of Technology, Saeki-ku, Hiroshima 731-5193, Japan b Department of Mathematics, Faculty of Science, Shizuoka University, Suruga-ku, Shizuoka 422-8529, Japan |
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Abstract: | By X(n), n?1, we denote the n-th symmetric hyperspace of a metric space X as the space of non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric. In this paper we shall describe the n-th symmetric hyperspace S1(n) as a compactification of an open cone over ΣDn−2, here Dn−2 is the higher-dimensional dunce hat introduced by Andersen, Marjanovi? and Schori (1993) [2] if n is even, and Dn−2 has the homotopy type of Sn−2 if n is odd (see Andersen et al. (1993) [2]). Then we can determine the homotopy type of S1(n) and detect several topological properties of S1(n). |
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Keywords: | Symmetric hyperspace Symmetric product Compactification Dunce hat |
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