Some remarks on hyperspaces

Abstract

1. The purpose of this paper is to answer a question of R. Schori [3] and to provide simpler arguments for some generalizations of Schori's results. If X is a metric space, the hyperspace of X, denoted 2X, is the space of all nonvoid closed subsets of X with the usual Hausdorff metric. The n-fold (n'Sil) symmetric product (Borsuk-Ulam [l]) of X, denoted X(n), is the subspace of 2X consisting of all elements with g« points. Let denote the closed unit interval, 7 the w-cube and 7°° the Hilbert cube. Let S(X) denote the subspace of 2X consisting of all continua. In [3] R. Schori shows that for w^l and a= °°, 1, 2, • • • , Ia(n) contains Ia as a factor; that is, I(n) is homeomorphic to Y XIa lor some space Y. Let 7°° denote another copy of the Hilbert cube with J=[ — 1, l] and let R be the equivalence relation on J°° defined by identifying each x = (xi, x2, • • • ) with —x = (-*i, -Xi, ■ • ■).