Brouwer fixed point theorem
TheoremLet be the closed unit ball in. Any continuous function
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 ora triangle
.
- Compactness counts (a)
The theorem is not true if we drop a point from the interior of B. For example, the map has the single fixed point at ; 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 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 |