Dirichlet conditions: Difference between revisions
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{{Orphan|date=February 2013}} | |||
In [[mathematics]], a [[Hausdorff space]] ''X'' is called a '''fixed-point space''' if every [[continuous function]] <math>f:X\rightarrow X</math> has a [[fixed point (mathematics)|fixed point]]. | |||
For example, any closed interval [a,b] in <math>\mathbb R</math> is a fixed point space, and it can be proved from the intermediate value property of real continuous function. The [[open interval]] (''a'', ''b''), however, is not a fixed point space. To see it, consider the function | |||
<math>f(x) = a + \frac{1}{b-a}\cdot(x-a)^2</math>, for example. | |||
Any [[linearly ordered]] space that is connected and has a top and a bottom element is a fixed point space. | |||
Note that, in the definition, we could easily have disposed of the condition that the space is Hausdorff. | |||
==References== | |||
* Vasile I. Istratescu, ''Fixed Point Theory, An Introduction'', D. Reidel, the Netherlands (1981). ISBN 90-277-1224-7 | |||
* Andrzej Granas and [[James Dugundji]], ''Fixed Point Theory'' (2003) Springer-Verlag, New York, ISBN 0-387-00173-5 | |||
* William A. Kirk and Brailey Sims, ''Handbook of Metric Fixed Point Theory'' (2001), Kluwer Academic, London ISBN 0-7923-7073-2 | |||
[[Category:Fixed points (mathematics)]] | |||
[[Category:Topology]] | |||
[[Category:Topological spaces]] | |||
{{mathanalysis-stub}} | |||
Revision as of 20:52, 22 October 2013
In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function has a fixed point.
For example, any closed interval [a,b] in is a fixed point space, and it can be proved from the intermediate value property of real continuous function. The open interval (a, b), however, is not a fixed point space. To see it, consider the function , for example.
Any linearly ordered space that is connected and has a top and a bottom element is a fixed point space.
Note that, in the definition, we could easily have disposed of the condition that the space is Hausdorff.
References
- Vasile I. Istratescu, Fixed Point Theory, An Introduction, D. Reidel, the Netherlands (1981). ISBN 90-277-1224-7
- Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5
- William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London ISBN 0-7923-7073-2