Fixed point space

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

For example, any closed interval [a,b] in 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 + 1 b − a ⋅ ( 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.