Brouwer fixed point theorem

Theorem Let 𝐁={xn:x1} be the closed unit ballPlanetmathPlanetmath in n. Any continuous functionMathworldPlanetmathPlanetmath f:𝐁𝐁 has a fixed pointPlanetmathPlanetmath.


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 triangleMathworldPlanetmath.

Compactness counts (a)

The theorem is not true if we drop a point from the interior of B. For example, the map f(x)=12x has the single fixed point at 0; dropping it from the domain yields a map with no fixed points (

Compactness counts (b)

The theorem is not true for an open disk. For instance, the map f(x)=12x+(12,0,,0) has its single fixed point on the boundary of B.

Title Brouwer fixed point theoremMathworldPlanetmath
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