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Transactions of the American Mathematical Society

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

 

 

Monotone decompositions of continua not separated by any subcontinua


Author: Eldon J. Vought
Journal: Trans. Amer. Math. Soc. 192 (1974), 67-78
MSC: Primary 54F20
DOI: https://doi.org/10.1090/S0002-9947-1974-0341438-X
MathSciNet review: 0341438
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Abstract: Let M be a compact, metric continuum that is separated by no subcontinuum. If such a continuum has a monotone, upper semicontinuous decomposition, the elements of which have void interior and for which the quotient space is a simple closed curve, then it is said to be of type $ {\text{A}}'$. It is proved that a bounded plane continuum is of type $ {\text{A'}}$ if and only if M contains no indecomposable subcontinuum with nonvoid interior. In $ {E^3}$ this condition is not sufficient and an example is given to illustrate this. However, it is shown that if M is hereditarily decomposable then M is of type $ {\text{A}}'$. Next, a condition is given that characterizes continua of type $ {\text{A'}}$. Also the structure of the elements in the decomposition of a continuum of type $ {\text{A'}}$ is discussed and the decomposition is shown to be unique. Finally, some consequences of these results and some remarks are given.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0341438-X
Keywords: Indecomposable continuum, continuum of type $ {\text{A'}}$, monotone decomposition, upper semicontinuous decomposition, core decomposition, hereditarily decomposable subcontinuum, simple closed curve
Article copyright: © Copyright 1974 American Mathematical Society