On a problem of Borsuk and Ulam
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- by Alexander Y. Gordon PDF
- Proc. Amer. Math. Soc. 137 (2009), 2135-2137 Request permission
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 đť‘‘-dimensional
torus.
References
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- Karol Borsuk, Über gewisse Invarianten der $\varepsilon$-Abbildungen, Math. Ann. 108 (1933), no. 1, 311–318 (German). MR 1512851, DOI 10.1007/BF01452839
- M. K. Fort Jr., A theorem about topological $n$-cells, Proc. Amer. Math. Soc. 5 (1954), 456–459. MR 62434, DOI 10.1090/S0002-9939-1954-0062434-9
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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
- MR Author ID: 239917
- Email: aygordon@uncc.edu
- Received by editor(s): December 27, 2007
- Published electronically: January 13, 2009
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2135-2137
- MSC (2000): Primary 57N99
- DOI: https://doi.org/10.1090/S0002-9939-09-09720-2
- MathSciNet review: 2480295