# fixed point property

Let $X$ be a topological space. If every continuous function $f\colon X\to X$ has a fixed 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 $h\colon X\to Y$ is a homeomorphism between topological spaces $X$ and $Y$, and $X$ has the fixed point property, and $f\colon Y\to Y$ 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

7. 7.

Brouwer’s Fixed Point Theorem states that in $\mathbb{R}^{n}$, 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. 8.

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

9. 9.

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

10. 10.

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

## Properties

1. 1.

Any topological space with the fixed point property is connected (http://planetmath.org/AnyTopologicalSpaceWithTheFixedPointPropertyIsConnected) and $\operatorname{T}_{0}$ (http://planetmath.org/T0Space).

2. 2.

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

3. 3.

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

## References

• 1 G. L. Naber, Topological methods in Euclidean spaces, Cambridge University Press, 1980.
• 2 G. J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
• 3 L. E. Ward, Topology, An Outline for a First Course, Marcel Dekker, Inc., 1972.
• 4 Charles Hagopian, The Fixed-Point Property for simply-connected plane continua, Trans. Amer. Math. Soc. 348 (1996) 4525–4548.
Title fixed point property FixedPointProperty 2013-03-22 13:56:32 2013-03-22 13:56:32 yark (2760) yark (2760) 20 yark (2760) Definition msc 55M20 msc 54H25 msc 47H10 fixed-point property FixedPoint