A Non-standard Version of the Borsuk–Ulam Theorem
Volume 53 / 2005
Bulletin Polish Acad. Sci. Math. 53 (2005), 111-119
MSC: Primary 54H25; Secondary 55M20, 47H09.
DOI: 10.4064/ba53-1-10
Abstract
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.
