Brouwer fixed-point theorem: Difference between revisions
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===In the language of spheres=== | ===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, for any natural number <math>n</math>, 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=== | ===In the language of simplices=== | ||
Any continuous map from the standard <math>n</math>-simplex, to itself, has a fixed point. | Any continuous map from the standard <math>n</math>-simplex, to itself has a fixed point. | ||
==Particular cases== | |||
===Case <math>n = 1</math>=== | |||
This says that any continuous map from the [[closed unit interval]] <math>[0,1]</math> to itself has a fixed point. This particular case is often proved as a consequence of the intermediate value theorem for continuous real-valued functions. Specifically, if <math>f:[0,1] \to [0,1]</math> is the function, then the function <math>g(x) := f(x) - x</math> crosses over from a non-positive to a nonnegative function and hence must be zero for some intermediate value of <math>x</math>. | |||
==Facts used== | ==Facts used== | ||
# [[uses::No-retraction theorem]] | # [[uses::No-retraction theorem]]: This states that there does not exist a continuous [[retraction]] from <math>D^n</math> to <math>S^{n-1}</math>, i.e., there is no continuous map from <math>D^n</math> to <math>S^{n-1} = \partial D^n</math> that restricts to the identity map on <math>S^{n-1}</math>. | ||
==Proof== | ==Proof== |
Latest revision as of 04:07, 24 December 2010
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, for any natural number , if denotes the spherical disc in , any continuous map must have a point such that .
In the language of simplices
Any continuous map from the standard -simplex, to itself has a fixed point.
Particular cases
Case
This says that any continuous map from the closed unit interval to itself has a fixed point. This particular case is often proved as a consequence of the intermediate value theorem for continuous real-valued functions. Specifically, if is the function, then the function crosses over from a non-positive to a nonnegative function and hence must be zero for some intermediate value of .
Facts used
- No-retraction theorem: This states that there does not exist a continuous retraction from to , i.e., there is no continuous map from to that restricts to the identity map on .
Proof
The Brouwer fixed-point theorem follows easily from the no-retraction theorem. Suppose is a continuous map with no fixed points. Define a map , that sends to the unique point on that is colllinear with and in such a way that lies between that point and . We can see that:
- Since is never equal to , and is inside the unit disc, is well-defined throughout
- is continuous
- is a retraction because it fixes every point on