Brouwer fixed-point theorem - Revision history

Vipul at 04:07, 24 December 2010

2010-12-24T04:07:44Z

← Older revision		Revision as of 04:07, 24 December 2010
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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==

Vipul at 04:04, 24 December 2010

2010-12-24T04:04:27Z

← Older revision		Revision as of 04:04, 24 December 2010
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	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.

	==~~Relation with other results~~==		==Facts used==

	~~===Results used in proving this===~~		# [[uses::No-retraction theorem]]

	* [[No-retraction theorem]]

	==Proof==		==Proof==

Vipul: 5 revisions

2008-05-11T19:32:29Z

5 revisions

← Older revision	Revision as of 19:32, 11 May 2008
(No difference)

Vipul at 04:23, 23 May 2007

2007-05-23T04:23:40Z

← Older revision		Revision as of 04:23, 23 May 2007
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	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.

			==Relation with other results==

			===Results used in proving this===

			* [[No-retraction theorem]]

			==Proof==

			The Brouwer fixed-point theorem follows easily from the [[no-retraction theorem]]. Suppose <math>f:D^n \to D^n</math> is a continuous map with no fixed points. Define a map <math>g:D^n \to S^{n-1}</math>, that sends <math>x \in D^n</math> to the unique point on <math>S^{n-1}</math> that is colllinear with <math>x</math> and <math>f(x)</math> in such a way that <math>x</math> lies between that point and <math>f(x)</math>. We can see that:

			* Since <math>f(x)</math> is never equal to <math>x</math>, and <math>x</math> is inside the unit disc, <math>g</math> is well-defined throughout <math>D^n</math>
			* <math>g</math> is continuous
			* <math>g</math> is a retraction because it fixes every point on <math>S^{n-1}</math>

Vipul: /* In the language of spheres */

2007-05-23T04:19:18Z

In the language of spheres

← Older revision		Revision as of 04:19, 23 May 2007
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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, 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.

Vipul at 04:19, 23 May 2007

2007-05-23T04:19:05Z

← Older revision		Revision as of 04:19, 23 May 2007
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	===In the language of simplices===		===In the language of simplices===

	Any continuous map from ~~thethe~~ 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.

Vipul at 04:18, 23 May 2007

2007-05-23T04:18:58Z

← Older revision		Revision as of 04:18, 23 May 2007
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	===In the language of spheres===		===In the language of spheres===

	Any continuous map from a ~~sphere~~ to itself must have a fixed point. In other words, if <math>S^n</math> denotes the <math>n</math>~~-dimensional sphere~~, any continuous map <math>f:S^n \to S^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, 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 boundary of the~~ standard <math>n</math>-simplex, to itself, has a fixed point.		Any continuous map from thethe standard <math>n</math>-simplex, to itself, has a fixed point.

Vipul at 04:14, 23 May 2007

2007-05-23T04:14:55Z

New page

{{theorem about spheres}}

==Statement==

===In the language of spheres===

Any continuous map from a sphere to itself must have a fixed point. In other words, if <math>S^n</math> denotes the <math>n</math>-dimensional sphere, any continuous map <math>f:S^n \to S^n</math> must have a point <math>x</math> such that <math>f(x) = x</math>.

===In the language of simplices===

Any continuous map from the boundary of the standard <math>n</math>-simplex, to itself, has a fixed point.