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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An imbedding problem


Authors: J. W. Cannon and S. G. Wayment
Journal: Proc. Amer. Math. Soc. 25 (1970), 566-570
MSC: Primary 54.78
DOI: https://doi.org/10.1090/S0002-9939-1970-0259875-2
MathSciNet review: 0259875
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Abstract: If $H$ is an uncountable collection of pairwise disjoint continua in ${E^n}$, each homeomorphic to $M$, then there exists a sequence from $H$ converging homeomorphically to an element of $H$. In the present paper the authors show that if $\{ {M_i}\}$ is a sequence of continua in ${E^n}$ which converges homeomorphically to ${M_0}$ and such that for each $i,{M_i}$ and ${M_0}$ are disjoint and equivalently imbedded, then there exists an uncountable collection $H$ of pairwise disjoint continua in ${E^n}$, each homeomorphic to $M$. For $n = 2,\;3$, and $n \geqq 5$ it is shown that one cannot guarantee that the elements of $H$ have the same imbedding as ${M_0}$.


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Keywords: Homeomorphic convergence, equivalently imbedded, <IMG WIDTH="14" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\epsilon$">-homeomorphism, uncountable collection of continua
Article copyright: © Copyright 1970 American Mathematical Society