On a theorem of Hopf

Abstract

It is proved that for every continuous mapping $f:S^n\to R^n$ and $\delta ,0<\delta <2$, there exists a paire of points $x,y$ with $||x-y||=\delta$ and $f(x)=f(y)$. The well-known theorem of Hopf is deduced from the above proposition: for every closed cover $F_1,F_2,..,F_{n+1}$ of the $n$-sphere $S^n$ and for every $\delta ,0<\delta <2$, there exists a triple $x,y,F_i$ such that $||x-y||=\delta$ and $x,y\in F_i$