On continua comparable with all continua (Q1807589)

Continua \(X\) and \(Y\) are said to be comparable provided \(Y\) is a continuous image of \(X\) or \(X\) is a continuous image of \(Y\). Let \(L\) denote a harmonic fan \(H\) with a sequence of smaller and smaller harmonic fans attached to the endpoints of \(H\). The authors prove the following Theorem. For any metric continuum \(X\) at least one of the following conditions is satisfied: (a) \(X\) is incomparable with some Waraszkiewicz spiral; (b) \(X\) is incomparable with \(L\); (c) \(X\) is a continuous image of the harmonic fan \(H\). Corollary. A metric continuum \(X\) is comparable with every continuum if and only if \(X\) is a continuous image of the harmonic fan.