Fixed-point space

In mathematics, a Hausdorff space "X" is called a fixed-point space if every continuous function formula_1 has a fixed point.
For example, any closed interval [a,b] in formula_2 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 
formula_3, 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.