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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
DOI: https://doi.org/10.1090/S0002-9939-1986-0826508-5
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.


References [Enhancements On Off] (What's this?)

  • [1] R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653-663. MR 0043450 (13:265a)
  • [2] H. Cook, Continua which admit only the identity mapping onto nondegenerate subcontinua, Fund. Math. 60 (1967), 241-249. MR 0220249 (36:3315)
  • [3] -, Tree-likeness of dendroids and $ \lambda $-dendroids, Fund. Math. 68 (1970), 18-22.
  • [4] A. Emeryk and Z. Horbanowicz, On atomic mappings, Colloq. Math. 27 (1973), 49-55. MR 0326644 (48:4987)
  • [5] S. D. Illiadis, On classification of hereditarily decomposable continua, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 29 (1974), 60-65. MR 0367944 (51:4186)
  • [6] J. Krasinkiewicz and P. Minc, Nonexistence of universal continua for certain classes of curves, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 24 (1976), 733-741. MR 0431109 (55:4111)
  • [7] K. Kuratowski, Topology, Vol. II, Academic Press, New York, and Polish Scientific Publishers, Warsaw, 1968. MR 0259835 (41:4467)
  • [8] E. S. Thomas, Jr., Monotone decompositions of irreducible continua, Rozprawy Mat. 50 (1966), 74 pages. MR 0196721 (33:4907)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0826508-5
Keywords: $ \lambda $-dendroid, chainable, hereditarily decomposable, tranche
Article copyright: © Copyright 1986 American Mathematical Society

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