fixed point property
Let $X$ be a topological space^{}. If every continuous function^{} $f:X\to X$ 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 $h:X\to Y$ is a homeomorphism between topological spaces $X$ 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 $\mathbb{R}$ 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 $\mathbb{R}{\mathbb{P}}^{2n}$ has the fixed point property.

9.
Every simplyconnected plane continuum has the fixedpoint 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 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 $f:X\to X$ 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 FixedPoint Property for simplyconnected plane continua, Trans. Amer. Math. Soc. 348 (1996) 4525–4548.
Title  fixed point property 

Canonical name  FixedPointProperty 
Date of creation  20130322 13:56:32 
Last modified on  20130322 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  fixedpoint property 
Related topic  FixedPoint 