Misplaced Pages

Fixed-point property: Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editNext edit →Content deleted Content addedVisualWikitext
Revision as of 22:19, 10 September 2007 editPaulTanenbaum (talk | contribs)Extended confirmed users2,391 edits Added order theoretic sense← Previous edit Revision as of 00:21, 11 September 2007 edit undoPaulTanenbaum (talk | contribs)Extended confirmed users2,391 editsm References: spelling errorNext edit →
Line 17: Line 17:


== References == == References ==
*{{cite book | first = Bernd | last = Schröder | title = Ordered Sets | publisher = Birkhâuser Boston | year = 2002}} *{{cite book | first = Bernd | last = Schröder | title = Ordered Sets | publisher = Birkhäuser Boston | year = 2002}}


{{topology-stub}} {{topology-stub}}

Revision as of 00:21, 11 September 2007

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.

Properties

A retract of a space with the fixed point property also has the fixed point property.

A product of spaces with the fixed point property also has the fixed point property.

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 dimension has the fixed point property by the Brouwer fixed point theorem.


References

  • Schröder, Bernd (2002). Ordered Sets. Birkhäuser Boston.
Stub icon

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

Categories: