The structure of Whitney blocks

Abstract

Let X be a metric continuum and C(X) the hyperspace of subcontinua of X. A Whitney block is a set of the form μ−1([s,t]), where μ:C(X)→[0,1] is a Whitney map and 0≤s<t<1. In this paper we study continua for which Whitney blocks are homeomorphic to X×[0,1]. We characterize an arc, a simple closed curve and simple n-ods in terms of Whitney blocks. We also show that if X is arc-like, then each Whitney block for C(X) is 2-cell-like; and if X is circle-like, then each Whitney block of the form μ−1([0,t]) is ring-like.