Branchpoint Covering Theorems for Confluent and Weakly Confluent Maps

A branchpoint of a compactum X is a point which is the vertex of a simple triod in X. A surjective map /: X -» 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 % of maps on a class of compacta & covers the branchpoints of its image, then it is said that the branchpoint covering property holds for ff on 0. 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. By a branchpoint of a compactum (i.e., compact metric space) X we mean a point p of X which is the vertex of a simple triod lying in X. Given a surjective map f: X —> Y between compacta we will say that / covers the branchpoints of Y provided each branchpoint of Y is the image under / of a branchpoint of X. Of course, in general, / will not cover the branchpoints of its image, but if it does whenever / is required to be in a particular class ¥ of maps and A" in a particular class G of compacta, then we will say that the branchpoint covering property holds for '$ on G. It is an immediate consequence of Whyburn's classical theorem on the lifting of dendrites under light open maps [8,p.l88] that the branchpoint covering property holds for light open maps on compacta. In this paper we prove that the branchpoint covering property holds more generally for confluent and weakly confluent maps on certain classes of compacta. As a corollary we show that the weakly confluent image of a graph is a graph, thus generalizing a theorem of Whyburn [8,p. 182]. I. Confluent maps on hereditarily arcwise connected compacta. Following Charatonik [1], we define a surjective map/: X —> Y to be confluent if for each subcontinuum A of Y, each component of f~l(A) is mapped by/onto A. It follows from a theorem of Whyburn [8,p.l48] that open maps on compacta are confluent. Our first theorem shows that the branchpoint covering property holds for confluent maps on hereditarily arcwise connected compacta (i.e. compacta in which each subcontinuum is arcwise connected). Received by the editors March 21, 1973 and, in revised form, October 16, 1974. AMS (MOS) subject classifications (1970). Primary 54C10; Secondary 54F50.