Models for Cp(X) for atriodic continua

A characterization of Cp(X), the family of subcontinua of X containing a fixed point of X, when X is an atriodic continuum, is given as follows. Assume that Z is a continuum and consider the following three conditions: (1) Z is a planar absolute retract; (2) cut points of Z have component number two; (3) any true cyclic element of Z contains at most two cut points of Z. If X is an atriodic continuum and p∈X, then Cp(X) satisfies (1)–(3) and, conversely, if Z satisfies (1)–(3), then there exist an arc-like continuum (hence, atriodic) X and a point p∈X such that Cp(X) is homeomorphic to Z.