proceedings of the american mathematical society Volume 106, Number 2, June 1989 ON TREE-LIKE CONTINUA WHICH ARE HOMOGENEOUS WITH RESPECT TO CONFLUENT LIGHT MAPPINGS PAWEL KRUPSKI (Communicated by James E. West) Abstract. If A" is a tree-like continuum with property K which is homoge- neous with respect to confluent light mappings, then X contains no two non- degenerate subcontinua with the one-point intersection. This is a generalization of C. L. Hagopian's result concerning homogeneous X . 1. Introduction A space A is homogeneous with respect to a class M of mappings if for every two points x, y e X there exists a mapping f e M of A onto A such that f(x) = y. Many results concerning the generalized homogeneity have been obtained in recent years and a special interest was given to generalize theroems on homoge- neous continua for some (larger than homeomorphisms) classes M of mappings (see [2]). There are known one-dimensional continua which are homogeneous with respect to open light mappings but are not homogeneous (e.g. the one-point union of two Menger universal curves [2, p. 588]). However, we don't know such an example of a tree-like continuum. It was observed in [5] that if a continuum A is homogeneous with respect to open mappings and each proper subcontinuum of A is an arc, then X is not tree-like. Moreover, J. R. Prajs has recently proved that A is a solenoid, [8]. In this paper we present an immediate proof that no tree-like continuum with the property of Kelley which is homogeneous with respect to confluent light mappings contains two nondegenerate subcontinua with the one-point intersec- tion. In particular, a tree-like continuum which is homogeneous with respect to open light mappings contains no two nondegenerate subcontinua with the one- point intersection. The theorem is a generalization of Hagopian's result about Received by the editors July 15, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F20; Secondary 54F50, 54C10. Key words and phrases. Continuum, homogeneous, tree-like, properly K , confluent mapping, Effros' theorem, outlet point. ©1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page 531 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use