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On a problem of Borsuk and Ulam


Author: Alexander Y. Gordon
Journal: Proc. Amer. Math. Soc. 137 (2009), 2135-2137
MSC (2000): Primary 57N99
DOI: https://doi.org/10.1090/S0002-9939-09-09720-2
Published electronically: January 13, 2009
MathSciNet review: 2480295
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Abstract: Borsuk and Ulam posed the following problem: For an arbitrary closed subset $ C$ of the $ d$-dimensional sphere, does there exist a sequence of homeomorphisms of the sphere such that the sequence of images of every point of the sphere converges to a point of $ C$ and each point of $ C$ is the limit of such a sequence? The answer is known to be positive, but the existing proof is complicated. We give a simple proof that extends to some other manifolds including the $ d$-dimensional torus.


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Additional Information

Alexander Y. Gordon
Affiliation: Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, North Carolina 28223
Email: aygordon@uncc.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09720-2
Keywords: Sequence of homeomorphisms; $d$-dimensional sphere
Received by editor(s): December 27, 2007
Published electronically: January 13, 2009
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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