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The depth of tranches in $ \lambda$-dendroids

Author: Lee Mohler
Journal: Proc. Amer. Math. Soc. 96 (1986), 715-720
MSC: Primary 54F50; Secondary 54B15, 54F20
MathSciNet review: 826508
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Abstract: According to the well-known theory of Kuratowski, any hereditarily decomposable chainable continuum admits a decomposition into tranches. These tranches are themselves chainable and thus admit decompositions into their own tranches. We may thus define nested sequences $ \{ {T_\alpha }\} $ of tranches-within-tranches, indexed by countable ordinals $ \alpha $, and finally terminating in a singleton set. E. S. Thomas, Jr. has asked whether, for a given continuum $ C$, there is a countable ordinal bound on the length of all such nests $ \{ {T_\alpha }\} $ in $ C$. We answer Thomas's question in the affirmative. By generalizing the definitions, we obtain the same result for $ \lambda $-dendroids. We also answer, for chainable continua, a related question of Illiadis.

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Keywords: $ \lambda $-dendroid, chainable, hereditarily decomposable, tranche
Article copyright: © Copyright 1986 American Mathematical Society

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