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On a problem of Borsuk and Ulam

Author(s): Alexander Y. Gordon
Journal: Proc. Amer. Math. Soc. 137 (2009), 2135-2137.
MSC (2000): Primary 57N99
Posted: January 13, 2009
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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.

References:

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Arnold, V. I., Ordinary differential equations, 3rd ed., Springer, Berlin, 1992. MR 1162307 (93b:34001)

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K. Borsuk, S. M. Ulam, Über gewisse Invarianten der $ \varepsilon$-Abbildungen, Math. Ann. 108 (1933), 311-318. MR 1512851

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M. K. Fort, Jr., A theorem about topological $ n$-cells, Proc. Amer. Math. Soc. 5 (1954), 456-459. MR 0062434 (15:978d)

4.
M. K. Fort, Jr., Sequences of homeomorphisms on the $ n$-sphere, Proc. Amer. Math. Soc. 12 (1961), 361-363. MR 0132538 (24:A2378)

5.
S. M. Ulam, A collection of mathematical problems, Interscience Publishers, New York, 1960. MR 0120127 (22:10884)

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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: 10.1090/S0002-9939-09-09720-2
PII: S 0002-9939(09)09720-2
Keywords: Sequence of homeomorphisms; $d$-dimensional sphere
Received by editor(s): December 27, 2007
Posted: January 13, 2009
Communicated by: Alexander N. Dranishnikov
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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