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In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function f : X → X {\displaystyle f:X\rightarrow X} has a fixed point.
For example, any closed interval in R {\displaystyle \mathbb {R} } is a fixed point space, and it can be proved from the intermediate value property of real continuous function. The open interval , however, is not a fixed point space. To see it, consider the function f = a + 1 b − a ⋅ 2 {\displaystyle f=a+{\frac {1}{b-a}}\cdot ^{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.