On a problem of Borsuk and Ulam

Gordon, Alexander

doi:10.1090/S0002-9939-09-09720-2

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 𝑑-dimensional

torus.

References

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