Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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
  • [2] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241–249. MR 0220249
  • [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
  • [5] S. D. Iliadis, The classification of hereditarily decomposable continua, Vestnik Moskov. Univ. Ser. I Mat. Meh. 29 (1974), no. 6, 60–65 (Russian, with English summary). MR 0367944
  • [6] J. Krasinkiewicz and P. Minc, Nonexistence of universal continua for certain classes of curves, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 9, 733–741 (English, with Russian summary). MR 0431109
  • [7] K. Kuratowski, Topology. Vol. II, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. MR 0259835
  • [8] E. S. Thomas Jr., Monotone decompositions of irreducible continua, Rozprawy Mat. 50 (1966), 74. MR 0196721

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54F50, 54B15, 54F20

Retrieve articles in all journals with MSC: 54F50, 54B15, 54F20


Additional Information

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