On the symmetric hyperspace of the circle

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).