Spheres, symmetric products, and quotient of hyperspaces of continua

Abstract

A continuum means a nonempty, compact and connected metric space. Given a continuum X, the symbols F_n(X) and C₁(X) 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 SF₁ⁿ(X) = F_n(X)/F₁(X) and C₁(X)/F₁(X) obtained by shrinking F₁(X) to a point in F_n(X) and C₁(X), respectively. In this paper, we study the continua X such that SF₁ⁿ(X) is homeomorphic to C₁(X)/F₁(X) and we analyze when the spaces F_n(X) and SF₁ⁿ(X) are homeomorphic to some sphere.