Fixed point space

In mathematics, a Hausdorff space "X" is called a fixed-point space if every continuous function $f:X\; ightarrow\; X$ 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