Confluent images of the hairy arc (Q1578895)

The hairy arc was introduced and characterized by \textit{J. M. Aarts} and \textit{L. G. Oversteegen} [Trans. Am. Math. Soc. 338, No. 2, 897-918 (1993; Zbl 0809.54034)] as a comb each of whose hairs is a two-sided limit of arcs. A confluent mapping \(f:X\to Y\) between continua \(X\) and \(Y\) was defined by the reviewer [Fundam. Math. 56, 213-220 (1964; Zbl 0134.18904)] as such that each component of the preimage of any subcontinuum \(Q\) of the range \(Y\) is mapped onto \(Q\) under \(f\). In the paper under review the authors investigate confluent images of the hairy arc. They introduce a class of smooth dendroids (named weak hairy arcs) which generalizes the hairy arc, and show that the confluent images of the hairy arc are contained in this class. It is also shown that if \(f:X\to Y\) is a confluent mapping defined on the hairy arc \(X\), then the properties of \(f\) of being: (a) open, (b) light, and (c) finite-to-one, are equivalent. The paper consists of seven sections: 1. Introduction. 2. Images of hairs. 3. Intersections of image hairs. 4. The geometry of branch points. 5. The weak hairy arc. 6. Open maps. 7. Examples.