Hierarchy of curves with weakly confluent maps
Tom 174 / 2023
Streszczenie
Given continua , Y and a class \mathcal F of maps between continua, define X\geq _{\mathcal F}Y if there exists an onto map f:X\rightarrow Y belonging to \mathcal F. A map f:X\rightarrow 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 \mathcal W of weakly confluent maps. We study the hierarchy of curves with respect to the partial order \leq _{\mathcal W}. Two continua X and Y are \mathcal W-equivalent provided that X\leq _{\mathcal W}Y and Y\leq _{\mathcal W}X. A continuum X is \mathcal W-isolated provided that the following implication holds: if Y is a continuum and X and Y are \mathcal W-equivalent, then X and Y are homeomorphic. Among other results, (a) we study how the class of dendrites with finite set of ramification points behaves under \leq _{\mathcal W}, (b) using \leq _{\mathcal W}, we compare dendrites with other curves, (c) we characterize \mathcal W-isolated finite graphs.
