Models for C_p(X) for atriodic continua

摘要

A characterization of C_p(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 C_p(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 C_p(X) is homeomorphic to Z.