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/