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Continuously homogeneous continua and their arc components


Author: Janusz R. Prajs
Journal: Proc. Amer. Math. Soc. 101 (1987), 533-540
MSC: Primary 54F20; Secondary 54F65
DOI: https://doi.org/10.1090/S0002-9939-1987-0908664-4
MathSciNet review: 908664
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Abstract: Let $ X$ be a continuously homogeneous Hausdorff continuum. We prove that if there is a sequence $ {A_1},{A_2}, \ldots $ of its arc components with $ X = {\text{c1}}{A_1} \cup {\text{c1}}{A_2} \cup \cdots $, and there is an arc component of $ X$ with nonempty interior, then $ X$ is arcwise connected. As consequences and applications we get: (1) if $ X$ is the countable union of arcwise connected continua, then $ X$ is arcwise connected; (2) if $ X$ is nondegenerate and metric, the number of its arc components is countable and it contains no simple triod, then it is either an arc or a simple closed curve; and, in particular, (3) an arc is the only nondegenerate continuously homogeneous arc-like metric continuum with countably many arc components.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0908664-4
Keywords: Continuous homogeneity, covering sequence, Hausdorff continuum, $ \mathcal{K}$-component, simple triod
Article copyright: © Copyright 1987 American Mathematical Society

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