Английская Википедия:Fixed-point space

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Шаблон:Short description Шаблон:No footnotes 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.

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). Шаблон:ISBN
• Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, Шаблон:ISBN
• William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London Шаблон:ISBN

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