Formal ethics: Difference between revisions
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A [[mathematics|mathematical]] object ''X'' has '''the fixed-point property''' if every suitably well-behaved [[mapping (mathematics)|mapping]] from ''X'' to itself has a [[fixed point (mathematics)|fixed point]]. The term is most commonly used to describe [[topological space]]s on which every [[continuous (mathematics)|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 (mathematics)|function]]) <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. | |||
==Examples== | |||
===The closed interval=== | |||
The [[closed interval]] [0,1] has the fixed point property: Let ''f'':[0,1] → [0,1] be a continuous 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 ''x''<sub>0</sub> with ''g''(''x''<sub>0</sub>) = 0, which is to say that ''f''(''x''<sub>0</sub>) − ''x''<sub>0</sub> = 0, and so ''x''<sub>0</sub> is a fixed point. | |||
The [[open interval]] does ''not'' have the fixed-point property. The mapping ''f''(''x'') = ''x''<sup>2</sup> 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]]. | |||
==Topology== | |||
A [[Deformation retract|retract]] ''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 topological space has the fixed-point property if and only if its identity map is [[universal map|universal]]. | |||
A [[product (topology)| 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. | |||
The FPP is a [[topological invariant]], i.e. is preserved by any [[homeomorphism]]. The FPP is also preserved by any [[retraction]]. | |||
According to [[Brouwer fixed point theorem]] every [[compact space|compact]] and [[convex set|convex]] subset of an [[euclidean space]] has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 [[Karol Borsuk|Borsuk]] asked whether compactness together with [[contractible space|contractibility]] could be a sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by [[Shin'ichi Kinoshita|Kinoshita]] who found an example of a compact contractible space without the FPP.<ref>Kinoshita, S. On Some Contractible Continua without Fixed Point Property. ''Fund. Math.'' '''40''' (1953), 96–98</ref> | |||
== References == | |||
<references/> | |||
*{{cite book | first = Norman Steenrod | last = Samuel Eilenberg | title = Foundations of Algebraic Topology | publisher = Princeton University Press | year = 1952}} | |||
*{{cite book | first = Bernd | last = Schröder | title = Ordered Sets | publisher = Birkhäuser Boston | year = 2002}} | |||
[[Category:Fixed points (mathematics)]] | |||
Revision as of 00:11, 7 January 2013
A mathematical object X has the fixed-point property if every suitably well-behaved mapping from X to itself has a fixed point. 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) 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 has a fixed point.
Examples
The closed interval
The closed interval [0,1] has the fixed point property: Let f:[0,1] → [0,1] be a continuous 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) = x2 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.
Topology
A retract A of a space X with the fixed-point property also has the fixed-point property. This is because if is a retraction and is any continuous function, then the composition (where is inclusion) has a fixed point. That is, there is such that . Since we have that and therefore
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.
The FPP is a topological invariant, i.e. is preserved by any homeomorphism. The FPP is also preserved by any retraction.
According to Brouwer fixed point theorem every compact and convex subset of an euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.[1]
References
- ↑ Kinoshita, S. On Some Contractible Continua without Fixed Point Property. Fund. Math. 40 (1953), 96–98
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