On a problem of Borsuk and Ulam

Author:
Alexander Y. Gordon

Journal:
Proc. Amer. Math. Soc. **137** (2009), 2135-2137

MSC (2000):
Primary 57N99

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 of the -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 and each point of 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 -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.