Brouwer fixed point theorem

Theorem Let $\text{𝐁}=\{x\in {ℝ}^n:\parallel x\parallel \le 1\}$ be the closed unit ball in ${ℝ}^n$. Any continuous function $f:\text{𝐁}\to \text{𝐁}$ 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)

The theorem is not true for an open disk. For instance, the map $f(\overrightarrow{x})=\frac{1}{2}\overrightarrow{x}+(\frac{1}{2},0,\mathrm{\dots },0)$ has its single fixed point on the boundary of B.

Title	Brouwer fixed point theorem
Canonical name	BrouwerFixedPointTheorem
Date of creation	2013-03-22 12:44:34
Last modified on	2013-03-22 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