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A characterization of continua that contain no $ n$-ods and no $ W$-sets

Author: Eldon J. Vought
Journal: Proc. Amer. Math. Soc. 109 (1990), 545-551
MSC: Primary 54F20; Secondary 54F25
MathSciNet review: 1012940
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Abstract: A proper, nondegenerate subcontinuum $ K$ of a continuum $ Y$ is said to be a $ W$-set if, for every continuum $ X$ and map $ f$ of $ X$ onto $ Y$, some subcontinuum of $ X$ is mapped by $ f$ onto $ K$. Jim Davis asked whether a simple closed curve is the only atriodic continuum that contains no $ W$-set. An affirmative answer is given to this question. The result follows as a corollary to the more general theorem that a continuum contains no $ n$-od and has no $ W$-set if and only if it is a graph in which every point is contained in a simple closed curve. Properties of this class of graphs are also described.

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Keywords: Continuum, $ W$-set, $ n$-od, atriodic, class $ W$, $ {\theta _n}$-continuum, graph, monotone upper-semicontinuous decomposition, quotient space
Article copyright: © Copyright 1990 American Mathematical Society

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