WEAKLY MONOTONE IMAGES OF SMOOTH DENDROIDS ARE SMOOTH

Abstract. A smooth dendroid is an arcwise connected, hereditarily unico-herent, metric continuum having a point p such that if ai- • a, then the unique arcs pai converge to pa. A mapping f from X onto Y is weakly mono-tone if, for each subcontinuum C of Y with nonvoid interior, each component of f- • (C) maps onto C. Maakowiak has shown that a weakly monotone image of a dendroid is a dendroid, and has asked if a weakly monotone image of a smooth dendroid is smooth. This paper gives a characterization of nonsmoothness for dendroids, and answers Ma•kowiak's question in the affirmative. 1. Introduction. Madkowiak proved in [5, page 724, Corollary 3.4] that a confluent image of a smooth dendroid is a smooth dendroid. He later proved [6, Theorem 3, page 179] that a weakly monotone image of a dendroid is a dendroid, and raised the questions [6, page 181] as to whether quasi-monotone or weakly monotone images of smooth dendroids are smoot