Rigidity of hyperspaces

Abstract

Given a metric continuum $X$, we consider the following hyperspaces of $X$: $2^{X}$, $C_{n}(X)$ and $F_{n}(X)$ ($n\in \mathbb{N}$). Let $F_{1}(X)=\{\{x\}:x\in X\}$. A hyperspace $K(X)$ of $X$ is said to be rigid, provided that for every homeomorphism $h:K(X)\rightarrow K(X)$, we have $h(F_{1}(X))=F_{1}(X)$. In this paper, we study conditions under which a continuum $X$ has a rigid hyperspace $C_{n}(X)$. Among others, we consider families of continua, such as dendroids, Peano continua, hereditarily indecomposable continua and smooth fans.