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.