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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cantor sets and homotopy connectedness of manifolds


Author: David G. Wright
Journal: Proc. Amer. Math. Soc. 50 (1975), 463-470
MSC: Primary 57A15
MathSciNet review: 0370594
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Abstract: We prove that a topological manifold $ M$ of dimension $ n$ is $ (n - 2)$-connected if each Cantor set in $ M$ is contained in an open $ n$-ball of $ M$. An immediate consequence is that a compact manifold $ N$ of dimension $ n(n \geq 5)$ is homeomorphic to the $ n$-sphere if and only if every Cantor set of $ N$ is contained in an open $ n$-ball of $ N$. This consequence generalizes a $ 3$-dimensional theorem of Doyle and Hocking.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0370594-9
PII: S 0002-9939(1975)0370594-9
Keywords: Cantor set, homotopy connected, engulfing, characterization of $ n$-sphere
Article copyright: © Copyright 1975 American Mathematical Society