The hyperspaces C(p,X) for atriodic continua

Abstract

Let C(X) denote the hyperspace of subcontinua of a continuum X. For A is an element of C(X), define the hyperspace C(A, X) = {B is an element of C(X) : A subset of B}. We prove that nondegenerate Whitney levels of C(p, X) are arcs when X is an atriodic continuum and p is an element of X. The main result is a characterization of the hyperspaces C(p, X) for atriodic continua. Moreover, as a consequence of the characterization, we obtain that a continuum X is atriodic if and only if C(A, X) is planar for all A is an element of C(X)