Brouwer fixed point theorem
Theorem Let $\text{\mathbf{B}}=\{x\in {\mathbb{R}}^{n}:\parallel x\parallel \le 1\}$ be the closed unit ball^{} in ${\mathbb{R}}^{n}$. Any continuous function^{} $f:\text{\mathbf{B}}\to \text{\mathbf{B}}$ has a fixed point^{}.
Notes
 Shape is not important

The theorem also applies to anything homeomorphic to a closed disk, of course. In particular, we can replace B in the formulation with a square or a triangle^{}.
 Compactness counts (a)

The theorem is not true if we drop a point from the interior of B. For example, the map $f(\overrightarrow{x})=\frac{1}{2}\overrightarrow{x}$ has the single fixed point at $0$; dropping it from the domain yields a map with no fixed points (http://planetmath.org/FixedPoint).
 Compactness counts (b)
Title  Brouwer fixed point theorem^{} 
Canonical name  BrouwerFixedPointTheorem 
Date of creation  20130322 12:44:34 
Last modified on  20130322 12:44:34 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  7 
Author  mathcam (2727) 
Entry type  Theorem 
Classification  msc 55M20 
Classification  msc 54H25 
Classification  msc 47H10 
Related topic  FixedPoint 
Related topic  SchauderFixedPointTheorem 
Related topic  TychonoffFixedPointTheorem 
Related topic  KKMlemma 
Related topic  KKMLemma 