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 be a continuously homogeneous Hausdorff continuum. We prove that if there is a sequence of its arc components with , and there is an arc component of with nonempty interior, then is arcwise connected. As consequences and applications we get: (1) if is the countable union of arcwise connected continua, then is arcwise connected; (2) if 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,
-component,
simple triod

Article copyright:
© Copyright 1987
American Mathematical Society