Continuously homogeneous continua and their arc components
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- by Janusz R. Prajs PDF
- Proc. Amer. Math. Soc. 101 (1987), 533-540 Request permission
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.References
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Additional Information
- © Copyright 1987 American Mathematical Society
- 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