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Proceedings of the American Mathematical Society

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

 

 

Scrawny Cantor sets are not definable by tori


Author: Amy Babich
Journal: Proc. Amer. Math. Soc. 115 (1992), 829-836
MSC: Primary 57M30; Secondary 28A05, 54G15, 57N12
DOI: https://doi.org/10.1090/S0002-9939-1992-1106178-4
MathSciNet review: 1106178
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Abstract: We define a Cantor set $ C$ in $ {{\mathbf{R}}^3}$ to be scrawny if for each $ p \in C$ and each $ \varepsilon > 0$ there is a $ \delta > 0$ such that for each map $ f:{S^1} \to \operatorname{Int}\,B(p,\delta ) - C$ there is a map $ F:{D^2} \to \operatorname{Int}{\mkern 1mu} B(p,\varepsilon )$ such that $ F\vert\partial {D^2} = f$ and $ {F^{ - 1}}(C)$ is finite. We show the existence and explore some of the properties of wild scrawny Cantor sets in $ {{\mathbf{R}}^3}$. We prove, among other things, that wild scrawny Cantor sets in $ {{\mathbf{R}}^3}$ are not definable by solid tori.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1106178-4
Article copyright: © Copyright 1992 American Mathematical Society