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 S 1 ( n ) as a compactification of an open cone over Σ D 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 S 1 ( n ) and detect several topological properties of S 1 ( n ) .