Monotone retractions and depth of continua. (English) Zbl 0817.54011

A continuum which is both hereditarily decomposable and hereditarily unicoherent is called a \(\lambda\)-dendroid. The depth of a \(\lambda\)- dendroid, \(X\), is defined as the minimum ordinal number \(\gamma\) such that the order type of each normal sequence of subcontinua of \(X\) is not greater than \(\gamma\). For additional information related to this concept the reader is referred to works by S. D. Iliadis [Moscow Univ. Math. Bull. 29, No. 5/6, 94-99 (1974); translation from Vestn. Mosk. Univ. Ser. I 29, No. 6, 60-65 (1974; Zbl 0294.54030)], L. Mohler [Proc. Am. Math. Soc. 96, 715-720 (1986; Zbl 0587.54054)], and the second author [Houston J. Math. 12, No. 4, 587-599 (1986; Zbl 0713.54038)]. While depth is preserved by homeomorphisms on continua, and by local homeomorphisms of \(\lambda\)-dendroids, an open mapping can change the depth even though every local homeomorphism is an open mapping. In this paper it is shown that for every two countable ordinals \(\alpha\) and \(\beta\) with \(\alpha>\beta\) there exist \(\lambda\)-dendroids \(X\) and \(Y\) whose depths are \(\alpha\) and \(\beta\) respectively, and a monotone retraction from \(X\) onto \(Y\). Moreover, it is shown that the continua \(X\) onto \(Y\) can be either both arclike or both fans. Several problems and open questions are presented.


54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54F15 Continua and generalizations
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