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Proceedings of the American Mathematical Society

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

 
 

 

Monotone decompositions of Hausdorff continua


Author: Eldon J. Vought
Journal: Proc. Amer. Math. Soc. 56 (1976), 371-376
MSC: Primary 54F15
DOI: https://doi.org/10.1090/S0002-9939-1976-0410693-7
MathSciNet review: 0410693
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Abstract: A monotone, upper semicontinuous decomposition of a compact, Hausdorff continuum is admissible if the layers (tranches) of the irreducible subcontinua of $ M$ are contained in the elements of the decomposition. It is proved that the quotient space of an admissible decomposition is hereditarily arcwise connected and that every continuum $ M$ has a unique, minimal admissible decomposition $ \mathcal{A}$. For hereditarily unicoherent continua $ \mathcal{A}$ is also the unique, minimal decomposition with respect to the property of having an arcwise connected quotient space. A second monotone, upper semicontinuous decomposition $ \mathcal{G}$ is constructed for hereditarily unicoherent continua that is the unique minimal decomposition with respect to having a semiaposyndetic quotient space. Then $ \mathcal{G}$ refines $ \mathcal{G}$ and $ \mathcal{G}$ refines the unique, minimal decomposition $ \mathcal{L}$ of FitzGerald and Swingle with respect to the property of having a locally connected quotient space (for hereditarily unicoherent continua).


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DOI: https://doi.org/10.1090/S0002-9939-1976-0410693-7
Article copyright: © Copyright 1976 American Mathematical Society

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