In mathematics, a Hausdorff space X is called a fixed-point space if every homeomorphism 
has a fixed point.
For example, any closed interval [a,b] of the real number line is a fixed point space: every continuous function with f(a) > 0 and f(b) < 0 must cross the real axis somewhere in the interval. By contrast, the open interval (a,b) is not a fixed point space.
References
- Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, Holland (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