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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-1-10,
author = {Carlos Biasi and Denise de Mattos},
title = {A Non-standard Version of the Borsuk-Ulam Theorem},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {53},
year = {2005},
pages = {111-119},
zbl = {1114.54025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-1-10}
}
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/