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.

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

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 $\mathbb{R}$ do not have the fixed point property. For example, the map $x\mapsto x+1$ on $\mathbb{R}$ has no fixed point.
6.

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.

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.

For each $n\in\mathbb{N}$ , the real projective space $\mathbb{RP}^{2n}$ 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 $\operatorname{T}_{0}$ (http://planetmath.org/T0Space).
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.

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
Canonical name	FixedPointProperty
Date of creation	2013-03-22 13:56:32
Last modified on	2013-03-22 13:56:32
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	20
Author	yark (2760)
Entry type	Definition
Classification	msc 55M20
Classification	msc 54H25
Classification	msc 47H10
Synonym	fixed-point property
Related topic	FixedPoint