Branchpoint covering theorems for confluent and weakly confluent maps

Authors:
C. A. Eberhart, J. B. Fugate and G. R. Gordh

Journal:
Proc. Amer. Math. Soc. **55** (1976), 409-415

MSC:
Primary 54F50

MathSciNet review:
0410703

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Abstract: A branchpoint of a compactum is a point which is the vertex of a simple triod in . A surjective map is said to cover the branchpoints of if each branchpoint in is the image of some branchpoint in . 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 on . 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.

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DOI:
https://doi.org/10.1090/S0002-9939-1976-0410703-7

Keywords:
Branchpoint,
confluent map,
weakly confluent map,
hereditarily arcwise connected compactum,
hereditarily locally connected continuum,
graph

Article copyright:
© Copyright 1976
American Mathematical Society