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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Subcontinua with degenerate tranches in hereditarily decomposable continua


Authors: Lex G. Oversteegen and E. D. Tymchatyn
Journal: Trans. Amer. Math. Soc. 278 (1983), 717-724
MSC: Primary 54F20; Secondary 54F50
MathSciNet review: 701520
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Abstract: A hereditarily decomposable, irreducible, metric continuum $ M$ admits a mapping $ f$ onto $ [0,1]$ such that each $ {f^{ - 1}}(t)$ is a nowhere dense subcontinuum. The sets $ {f^{ - 1}}(t)$ are the tranches of $ M$ and $ {f^{ - 1}}(t)$ is a tranche of cohesion if $ t \in \{ 0,1\} $ or $ {f^{ - 1}}(t) = {\text{C1}}({f^{ - 1}}([0,t))) \cap {\text{C1}}\,({f^{ - 1}}((t,1]))$. The following answer a question of Mahavier and of E. S. Thomas, Jr.

Theorem. Every hereditarily decomposable continuum contains a subcontinuum with a degenerate tranche.

Corollary. If in an irreducible hereditarily decomposable continuum each tranche is nondegenerate then some tranche is not a tranche of cohesion.

The theorem answers a question of Nadler concerning arcwise accessibility in hyperspaces.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0701520-7
Keywords: Hereditarily decomposable continua, monotone decomposition into tranches
Article copyright: © Copyright 1983 American Mathematical Society