On the symmetric hyperspace of the circle (Q602030)

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 \(X\).