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On a characterization of $ W$-sets and the dimension of hyperspaces


Authors: J. Grispolakis and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 100 (1987), 557-563
MSC: Primary 54F20; Secondary 54B20, 54F45
DOI: https://doi.org/10.1090/S0002-9939-1987-0891163-6
MathSciNet review: 891163
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Abstract: A subcontinuum $ A$ of a continuum $ X$ is a $ W$-set if for each mapping $ f:Y \twoheadrightarrow X$ of an arbitrary continuum $ Y$ onto $ X$ there is a continuum in $ Y$ which is mapped by $ f$ onto $ A$. We characterize $ W$-sets in terms of accessibility by small continua. We localize several known results on continua all of whose subcontinua are $ W$-sets. Finally, we extend a result of J. T. Rogers by proving that if $ X$ is an atriodic continuum whose first Čech cohomology group is finitely generated then the hyperspace $ C(X)$ of subcontinua of $ X$ is two dimensional.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0891163-6
Keywords: Continua, weakly confluent mappings, dimension of hyperspaces, atriodic
Article copyright: © Copyright 1987 American Mathematical Society