Spheres, symmetric products, and quotient of hyperspaces of continua

. A continuum means a nonempty, compact and connected metric space. Given a continuum X , the symbols F n ð X Þ and C 1 ð X Þ denotes the hyperspace of all subsets of X with at most n points and the hyperspace of subcontinua of X , respectively. If n > 1, we consider the quotient spaces SF n 1 ð X Þ ¼ F n ð X Þ = F 1 ð X Þ and C 1 ð X Þ = F 1 ð X Þ obtained by shrinking F 1 ð X Þ to a point in F n ð X Þ and C 1 ð X Þ , re-spectively. In this paper, we study the continua X such that SF n 1 ð X Þ is homeomorphic to C 1 ð X Þ = F 1 ð X Þ and we analyze when the spaces F n ð X Þ and SF n 1 ð X Þ are homeomorphic to some sphere.