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Transactions of the American Mathematical Society

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



Monotone decompositions of $ \theta \sb{n}$-continua

Authors: E. E. Grace and Eldon J. Vought
Journal: Trans. Amer. Math. Soc. 263 (1981), 261-270
MSC: Primary 54F20; Secondary 54B15
MathSciNet review: 590423
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Abstract: We prove the following theorem for a compact, metric $ {\theta _n}$-continuum (i.e., a compact, connected, metric space that is not separated into more than $ n$ components by any subcontinuum). The continuum $ X$ admits a monotone, upper semicontinuous decomposition $ \mathfrak{D}$ such that the elements of $ \mathfrak{D}$ have void interiors and the quotient space $ X/\mathfrak{D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $ H$ of $ X$, the continuum $ T(H) = \{ x\vert$ if $ K$ is a subcontinuum of $ X$ and $ x \in {K^ \circ }$, then $ K \cap H \ne \emptyset \} $ is nowhere dense. The elements of the decomposition are characterized in terms of the set function $ T$. An example is given showing that the condition that requires $ T(x)$ to have void interior for all $ x \in X$ is not strong enough to guarantee the decomposition.

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Keywords: $ {\theta _n}$-continuum, monotone upper semicontinuous decomposition, quotient space, finite graph, aposyndetic set function $ T$, compact metric continuum
Article copyright: © Copyright 1981 American Mathematical Society