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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Disk-like products of $ \lambda $ connected continua. II


Author: Charles L. Hagopian
Journal: Proc. Amer. Math. Soc. 52 (1975), 479-484
MSC: Primary 54F20
DOI: https://doi.org/10.1090/S0002-9939-1975-0494000-9
MathSciNet review: 0494000
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Abstract: R. H. Bing [3] proved that every atriodic, hereditarily decomposable, hereditarily unicoherent continuum is arc-like. Using this theorem, the author [5] showed that $ \lambda $ connected continua $ X$ and $ Y$ are arc-like when the topological product $ X \times Y$ is disk-like. In this paper we consider products that have a more general mapping property. Suppose that $ X$ and $ Y$ are $ \lambda $ connected continua and that for each $ \varepsilon > 0$, there exists an $ \varepsilon $-map of $ X \times Y$ into the plane. Then $ X$ is either arc-like or circle-like. Furthermore, if $ X$ is circle-like, then $ Y$ is arc-like. Hence $ X \times Y$ is either disk-like or annulus-like.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0494000-9
Keywords: Chainable continua, snake-like continua, disk-like product, arc-like continua, lambda connectivity, hereditarily decomposable continua, arcwise connectivity, triod, unicoherence, circle-like continua, $ \varepsilon $-map into the plane
Article copyright: © Copyright 1975 American Mathematical Society