fixed point property

Let $X$ be a topological space.If every continuous function $f:X\to X$ has afixed point (http://planetmath.org/FixedPoint),then $X$ is said to have the fixed point property.

The fixed point property is obviously preserved under homeomorphisms.If $h:X\to Y$ is a homeomorphism between topologicalspaces $X$ and $Y$, and $X$ has the fixed point property,and $f:Y\to Y$ is continuous,then $h^{-1}\circ f\circ h$ has a fixed point $x\in X$,and $h(x)$ is a fixed point of $f$.

Examples

1. A space with only one point has the fixed point property.

2. A closed interval $[a,b]$ of $ℝ$ has the fixed point property.This can be seen using the mean value theorem. (http://planetmath.org/BrouwerFixedPointInOneDimension)

3. The extended real numbers have the fixed point property,as they are homeomorphic to $[0,1]$.

4. The topologist’s sine curve has the fixed point property.

5. The real numbers $ℝ$ do not have the fixed point property.For example, the map $x\mapsto x+1$ on $ℝ$ has no fixed point.

6. An open interval $(a,b)$ of $ℝ$ does not have the fixed point property.This follows since any such interval is homeomorphic to $ℝ$.Similarly, an open ball in ${ℝ}^n$ does not have the fixed point property.

7. Brouwer’s Fixed Point Theorem states that in ${ℝ}^n$,the closed unit ball with the subspace topology has the fixed point property.(Equivalently, ${[0,1]}^n$ has the fixed point property.)The Schauder Fixed Point Theorem generalizes this result further.

8. For each $n\in \mathbb{N}$, the real projective space $ℝ{\mathbb{P}}^{2n}$ has the fixed point property.

9. Every simply-connected plane continuum has the fixed-point property.

10. The Alexandroff–Urysohn square (also known as the Alexandroff square)has the fixed point property.

Properties

1. Any topological space with the fixed point property is connected (http://planetmath.org/AnyTopologicalSpaceWithTheFixedPointPropertyIsConnected)and ${\mathrm{T}}_0$ (http://planetmath.org/T0Space).

2. Suppose $X$ is a topological space with thefixed point property, and $Y$ is a retract of $X$. Then$Y$ has the fixed point property.

3. Suppose $X$ and $Y$ are topological spaces, and $X\times Y$ has thefixed point property. Then $X$ and $Y$ have the fixed point property.(Proof: If $f:X\to X$ is continuous,then $(x,y)\mapsto (f(x),y)$ is continuous, so $f$ has a fixed point.)