Rigidity of symmetric products

摘要

Given a metric continuum X, we consider the following hyperspaces of X: 2~X, C_n(X) and F_n(X)(n∈N). Let F_1(X) = {{x}: x∈ X}. A hyperspace K(X) of X is said to be rigid provided that for every homeomorphism h : K(X) → K(X) we have that h(F_1(X)) = F_1(X). In this paper we study under which conditions a continuum X has a rigid hyperspace F_n(X). Among others, we consider families of continua such as, dendroids, Peano continua, indecomposable arc continua (all their proper nondegenerate subcontinua are arcs), hereditarily indecomposable continua and smooth fans.