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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
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}$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0259875-2
Keywords: Homeomorphic convergence, equivalently imbedded, $ \epsilon $-homeomorphism, uncountable collection of continua
Article copyright: © Copyright 1970 American Mathematical Society