Hierarchy of curves with weakly confluent maps
Volume 174 / 2023
Abstract
Given continua $X$, $Y$ 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.
