Branchpoint covering theorems for confluent and weakly confluent maps

Abstract:

A branchpoint of a compactum $X$ is a point which is the vertex of a simple triod in $X$. A surjective map $f:X \to Y$ is said to cover the branchpoints of $Y$ if each branchpoint in $Y$ is the image of some branchpoint in $X$. If every map in a class $\mathcal {F}$ of maps on a class of compacta $\mathcal {C}$ covers the branchpoints of its image, then it is said that the branchpoint covering property holds for $\mathcal {F}$ on $\mathcal {C}$. According to Whyburn’s classical theorem on the lifting of dendrites, the branchpoint covering property holds for light open maps on arbitrary compacta. In this paper it is shown that the branchpoint covering property holds for (1) light confluent maps on arbitrary compacta, (2) confluent maps on hereditarily arcwise connected compacta, and (3) weakly confluent maps on hereditarily locally connected continua having closed sets of branchpoints. It follows that the weakly confluent image of a graph is a graph.