Fixed-point property: Difference between revisions

Browse history interactively ← Previous edit Next edit →Content deleted Content addedVisual WikitextInline

Revision as of 10:08, 10 January 2008 edit85.91.87.250 (talk)No edit summary← Previous edit		Revision as of 17:15, 3 October 2008 edit undoGandalfxviv (talk \| contribs)326 editsm added more rigorous definition and proof of the retract exampleNext edit →
Line 1:		Line 1:
	A ] object ''X'' has '''the fixed point property''' if every suitably well-behaved ] from ''X'' to itself has a ]. It is a special case of the ]. The term is most commonly used to describe ]s on which every ] mapping has a fixed point. But another use is in ], where a ] ''P'' is said to have the fixed point property if every ] on ''P'' has a fixed point.		A ] object ''X'' has '''the fixed point property''' if every suitably well-behaved ] from ''X'' to itself has a ]. It is a special case of the ]. The term is most commonly used to describe ]s on which every ] mapping has a fixed point. But another use is in ], where a ] ''P'' is said to have the fixed point property if every ] on ''P'' has a fixed point.

			==Definition==
			Let ''A'' be an object in the ] '''C'''. Then ''A'' has the ''fixed point property'' if every ] (i.e., every ]) <math>f:A\to A</math> has a fixed point.

			The most common usage is when '''C'''='''Top''' is the category of topological spaces. Then a topological space ''X'' has the fixed point property if every continuous map <math>f:X\to X</math> has a fixed point.

	==Properties==		==Properties==
			A ] ''A'' of a space ''X'' with the fixed point property also has the fixed point property. This is because if <math>r:X\to A</math> is a retraction and <math>f:A\to A</math> is any continuous function, then the composition <math>i\circ f\circ r:X\to X</math> (where <math>i:A\to X</math> is inclusion) has a fixed point. That is, there is <math>x\in A</math> such that <math>f\circ r(x)=x</math>. Since <math>x\in A</math> we have that <math>r(x)=x</math> and therefore <math>f(x)=x.</math>
	A ] of a space with the fixed point property also has the fixed point property.

	A topological space has the fixed point property if and only if its identity map is ].		A topological space has the fixed point property if and only if its identity map is ].

Revision as of 17:15, 3 October 2008

A mathematical object X has the fixed point property if every suitably well-behaved mapping from X to itself has a fixed point. It is a special case of the fixed morphism property. The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed point. But another use is in order theory, where a partially ordered set P is said to have the fixed point property if every increasing function on P has a fixed point.

Definition

Let A be an object in the concrete category C. Then A has the fixed point property if every morphism (i.e., every function) $f:A\to A$ has a fixed point.

The most common usage is when C=Top is the category of topological spaces. Then a topological space X has the fixed point property if every continuous map $f:X\to X$ has a fixed point.

Properties

A retract A of a space X with the fixed point property also has the fixed point property. This is because if $r:X\to A$ is a retraction and $f:A\to A$ is any continuous function, then the composition $i\circ f\circ r:X\to X$ (where $i:A\to X$ is inclusion) has a fixed point. That is, there is $x\in A$ such that $f\circ r(x)=x$ . Since $x\in A$ we have that $r(x)=x$ and therefore $f(x)=x.$

A topological space has the fixed point property if and only if its identity map is universal.

A product of spaces with the fixed point property in general fails to have the fixed point property even if one of the spaces is the closed real interval.

Examples

The closed interval

The closed interval has the fixed point property: Let f: → be a mapping. If f(0) = 0 or f(1) = 1, then our mapping has a fixed point at 0 or 1. If not, then f(0) > 0 and f(1) − 1 < 0. Thus the function g(x) = f(x) − x is a continuous real valued function which is positive at x = 0 and negative at x = 1. By the intermediate value theorem, there is some point x₀ with g(x₀) = 0, which is to say that f(x₀) − x₀ = 0, and so x₀ is a fixed point.

The open interval does not have the fixed point property. The mapping f(x) = x has no fixed point on the interval (0,1).

The closed disc

The closed interval is a special case of the closed disc, which in any finite dimension has the fixed point property by the Brouwer fixed point theorem.

References

Samuel Eilenberg, Norman Steenrod (1952). Foundations of Algebraic Topology. Princeton University Press.
Schröder, Bernd (2002). Ordered Sets. Birkhäuser Boston.

This topology-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: