Let $X$ be a topological space. If every continuous function $f\colon X\to X$ has a fixed point, 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 $\sR$ has the fixed point property. This can be seen using the mean value theorem.
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 $\sR$ do not have the fixed point property. For example, the map $x\mapsto x+1$ on $\sR$ has no fixed point.
6. An open interval $(a,b)$ of $\sR$ does not have the fixed point property. This follows since any such interval is homeomorphic to $\sR$ . Similarly, an open ball in $\sR^n$ does not have the fixed point property.
7. Brouwer's Fixed Point Theorem states that in $\sR^n$ , 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 $n\in\sN$ , the real projective space $\RP^{2n}$ 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 and $\operatorname{T}_0$ .
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.)

Bibliography

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.