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 σD^n-2, here D^n-2 is the higher-dimensional dunce hat introduced by Andersen, Marjanović and Schori (1993) [2] if n is even, and D^n-2 has the homotopy type of S^n-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).