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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
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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Keywords: Chainable continua, snake-like continua, disk-like product, arc-like continua, lambda connectivity, hereditarily decomposable continua, arcwise connectivity, triod, unicoherence, circle-like continua, <!– MATH $\varepsilon$ –> <IMG WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\varepsilon$">-map into the plane
Article copyright: © Copyright 1975 American Mathematical Society