A Non-standard Version of the Borsuk–Ulam Theorem

Tom 53 / 2005

Carlos Biasi, Denise de Mattos Bulletin Polish Acad. Sci. Math. 53 (2005), 111-119 MSC: Primary 54H25; Secondary 55M20, 47H09. DOI: 10.4064/ba53-1-10

Streszczenie

E. Pannwitz showed in 1952~%\cite{EP} that for any $n\geq 2$, there exist continuous maps $\varphi:S^{n}\to S^{n}$ and $f:S^{n}\to \mathbb{R}^{2}$ such that $f(x)\not = f(\varphi(x))$ for any $x\in S^{n}$. We prove that, under certain conditions, given continuous maps $\psi,\varphi:X\to X$ and $f:X\to \mathbb{R}^{2}$, although the existence of a point $x\in X$ such that $f(\psi(x))=f(\varphi(x))$ cannot always be assured, it is possible to establish an interesting relation between the points $f(\varphi \psi(x)), f(\varphi^{2}(x))$ and $f(\psi^{2}(x))$ when $f(\varphi(x))\not =f(\psi(x))$ for any $x\in X$, and a non-standard version of the Borsuk–Ulam theorem is obtained.

Autorzy

  • Carlos BiasiDepartamento de Matemática-ICMC
    Universidade de São Paulo
    Caixa Postal 668
    13560-970, São Carlos SP, Brazil
    e-mail
  • Denise de MattosDepartamento de Matemática–ICMC
    Universidade de São Paulo
    Caixa Postal 668
    13560-970, São Carlos SP, Brazil
    e-mail

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