A Non-standard Version of the Borsuk-Ulam Theorem
Carlos Biasi ; Denise de Mattos
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005), p. 111-119 / Harvested from The Polish Digital Mathematics Library

E. Pannwitz showed in 1952 that for any n ≥ 2, there exist continuous maps φ:Sⁿ→ Sⁿ and f:Sⁿ→ ℝ² such that f(x) ≠ f(φ(x)) for any x∈ Sⁿ. We prove that, under certain conditions, given continuous maps ψ,φ:X→ X and f:X→ ℝ², although the existence of a point x∈ X such that f(ψ(x)) = f(φ(x)) cannot always be assured, it is possible to establish an interesting relation between the points f(φ ψ(x)), f(φ²(x)) and f(ψ²(x)) when f(φ(x)) ≠ f(ψ(x)) for any x∈ X, and a non-standard version of the Borsuk-Ulam theorem is obtained.

Publié le : 2005-01-01
EUDML-ID : urn:eudml:doc:280821
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Carlos Biasi; Denise de Mattos. A Non-standard Version of the Borsuk-Ulam Theorem. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) pp. 111-119. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-1-10/