On the symmetric hyperspace of the circle
type: Item
instance of: scholarly article
title: On the symmetric hyperspace of the circle
zbMATH Open document ID: 1205.54018
author: Naotsugu Chinen
author: Akira Koyama
published in: Topology and its Applications
publication date: 2010-10-29T00:00:00Z
review text: Given a metric space \(X\), the \(n\)th symmetric hyperspace of \(X\), denoted by \(X(n)\), is the space of nonempty finite subsets of \(X\) with at most \(n\) elements equipped with the Hausdorff metric. Classical results of \textit{K. Borsuk} and \textit{S. Ulam} in [Bulletin A. M. S. 37, 875--882 (1931; JFM 57.1521.01)] claim that for \(n=1,2\), or 3, \({\mathbb I}(n)\) is homeomorphic to \({\mathbb I}^n\) and for \(n\geq 4\), \({\mathbb I}(n)\) cannot be embedded into \({\mathbb R}^n\), where \({\mathbb I} =[0,1]\) is the closed unit interval. In this paper, the authors study the \(n\)th symmetric hyperspace \({\mathbb S}^1(n)\) of the unit circle \({\mathbb S}^1 =\{(x,y) \in {\mathbb R}^2: x^2+y^2 =1\}\). The main result says that for \(n \geq 2\) there exists a closed subset \(R \subseteq {\mathbb S}^1(n)\) which is homeomorphic to \(D^{n-1}\) such that \({\mathbb S}^1(n) \setminus R\) is homeomorphic to the open cone over \(\Sigma D^{n-2}\). As corollaries, some classical results of \textit{W.-T. Wu} in [C. R. Acad. Sci., Paris 224, 1139--1141 (1947; Zbl 0029.07802)] are derived. It is also concluded that for any \(n\geq 4\), \({\mathbb S}^1(n)\) is not an \(n\)-manifold, and there exists no embedding of any orientable closed \(n\)-dimensional manifold into \({\mathbb S}^1(n)\). The authors also ask whether \({\mathbb S}^n\), \(n \geq 4\), is embeddable into the \(n\)th symmetric hyperspace \(X(n)\) of a curve \(X\).
reviewed by: Jiling Cao
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 5808685
zbMATH Keywords: symmetric hyperspace
zbMATH Keywords: symmetric product
zbMATH Keywords: compactification
zbMATH Keywords: dunce hat
MaRDI profile type: Publication
full work available at URL: j.topol.2010.07.012
OpenAlex ID: W2047554417
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DOI: 10.1016/J.TOPOL.2010.07.012
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description: scientific article; zbMATH DE number 5808685