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Brouwer fixed point theorem
From Mathematics Is A Science
Revision as of 13:35, 28 August 2015 by imported>WikiSysop (Redirected page to Euler and Lefschetz numbers#Fixed points)
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Theorem. Any continuous function on a closed unit ball has a fixed point.
In other words, suppose ${\bf B}$ $= \{ x \in {\bf R}^n: ||x|| \le 1 \} $ is the closed unit ball in ${\bf R}^n$ and $f: {\bf B}\to {\bf B}$ is a continuous function. Then there is $x\in {\bf B}$ such that $f(x)=x$.
The theorem holds if the ball is replaced with anything homeomorphic to it.
It is a particular case of the Lefschetz fixed point theorem.