Brouwer fixed-point theorem: Difference between revisions

Revision as of 04:19, 23 May 2007

This article describes a theorem about spheres

Statement

In the language of spheres

Any continuous map from a disc to itself must have a fixed point. In other words, if $D^{n}$ denotes the spherical disc in $\mathbb {R} ^{n}$ , any continuous map $f:D^{n}\to D^{n}$ must have a point $x$ such that $f(x)=x$ .

In the language of simplices

Any continuous map from the standard $n$ -simplex, to itself, has a fixed point.

@@ Line 5: / Line 5: @@
 ===In the language of spheres===
-Any continuous map from a disc to itself must have a fixed point. In other words, if <math>D^n</math> denotes the spherical disc in <math>\R&n</math>, any continuous map <math>f:D^n \to D^n</math> must have a point <math>x</math> such that <math>f(x) = x</math>.
+Any continuous map from a disc to itself must have a fixed point. In other words, if <math>D^n</math> denotes the spherical disc in <math>\R^n</math>, any continuous map <math>f:D^n \to D^n</math> must have a point <math>x</math> such that <math>f(x) = x</math>.
 ===In the language of simplices===
 Any continuous map from the standard <math>n</math>-simplex, to itself, has a fixed point.