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Monotone decompositions of $ \theta $-continua


Author: E. E. Grace
Journal: Trans. Amer. Math. Soc. 275 (1983), 287-295
MSC: Primary 54F20; Secondary 54B15, 54C60, 54E45, 54F65
DOI: https://doi.org/10.1090/S0002-9947-1983-0678350-8
MathSciNet review: 678350
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Abstract: A $ \theta $-continuum ( $ {\theta _n}$-continuum) is a compact, connected, metric space that is not separated into infinitely many (more than $ n$) components by any subcontinuum. The following results are among those proved. The first generalizes earlier joint work with E. J. Vought for $ {\theta _n}$-continua, and the second generalizes earlier work by Vought for $ {\theta _1}$-continua.

A $ \theta $-continuum $ X$ admits a monotone, upper semicontinuous decomposition $ \mathcal{D}$ such that the elements of $ \mathcal{D}$ have void interiors and the quotient space $ X/\mathcal{D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $ H$ of $ X$, the continuum $ T(H) = \{x \in X\vert$ if $ K$ is a subcontinuum of $ X$ and $ x$ is in the interior of $ K$, then $ K \cap H \ne \emptyset \} $ is nowhere dense. Also, if $ X$ satisfies this condition, then $ X$ is in fact a $ {\theta _n}$-continuum, for some natural number $ n$, and, for each natural number $ m$, $ X$ is a $ {\theta _m}$-continuum, if and only if $ X/\mathcal{D}$ is a $ {\theta _m}$-continuum.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0678350-8
Keywords: $ \theta $-continuum, $ {\theta _n}$-continuum, $ n$-od, monotone decomposition, upper semicontinuous decomposition, decomposition space, finite graph, aposyndetic set function $ T$, compact metric continuum, $ T$-closed set, condensation decomposition
Article copyright: © Copyright 1983 American Mathematical Society

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