Fixed-point space

In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function $f:X\rightarrow X$ has a fixed point.

For example, any closed interval [a,b] in $\mathbb {R}$ 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 $f(x)=a+{\frac {1}{b-a}}\cdot (x-a)^{2}$ , 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

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