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Proceedings of the American Mathematical Society

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

 
 

 

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