HIERARCHY OF CURVES WITH WEAKLY CONFLUENT MAPS

Given continua X,Y and a class F of maps between continua, define X >= Y-F if there exists an onto map f : X -> Y belonging to F. A map f : X -> Y is weakly confluent if for each subcontinuum B of Y, there exists a subcontinuum A of X such that f(A) =B. In this paper we consider the class W of weakly confluent maps. We study the hierarchy of curves with respect to the partial order <=(W). Two continua X and Y are W-equivalent provided that X <= WY and Y <=(W) X. A continuum X is W-isolated provided that the following implication holds: if Y is a continuum and X and Y are W-equivalent, then X and Yare homeomorphic. Among other results, (a) we study how the class of dendrites with finite set of ramification points behaves under <=(W),(b) using <=(W), we compare dendrites with other curves, (c) we characterize W-isolated finite graphs.