# Fixed-point property

A mathematical object *X* has **the fixed-point property** if every suitably well-behaved mapping from *X* to itself has a fixed point. The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed point. But another use is in order theory, where a partially ordered set *P* is said to have the fixed point property if every increasing function on *P* has a fixed point.

## Definition

Let *A* be an object in the concrete category **C**. Then *A* has the *fixed-point property* if every morphism (i.e., every function) has a fixed point.

The most common usage is when **C**=**Top** is the category of topological spaces. Then a topological space *X* has the fixed-point property if every continuous map has a fixed point.

## Examples

### Singletons

In the category of sets, the objects with the fixed-point property are precisely the singletons.

### The closed interval

The closed interval [0,1] has the fixed point property: Let *f*:[0,1] → [0,1] be a continuous mapping. If *f*(0) = 0 or *f*(1) = 1, then our mapping has a fixed point at 0 or 1. If not, then *f*(0) > 0 and *f*(1) − 1 < 0. Thus the function *g*(*x*) = *f*(*x*) − x is a continuous real valued function which is positive at *x* = 0 and negative at *x* = 1. By the intermediate value theorem, there is some point *x*_{0} with *g*(*x*_{0}) = 0, which is to say that *f*(*x*_{0}) − *x*_{0} = 0, and so *x*_{0} is a fixed point.

The open interval does *not* have the fixed-point property. The mapping *f*(*x*) = *x*^{2} has no fixed point on the interval (0,1).

### The closed disc

The closed interval is a special case of the closed disc, which in any finite dimension has the fixed-point property by the Brouwer fixed-point theorem.

## Topology

A retract *A* of a space *X* with the fixed-point property also has the fixed-point property. This is because if is a retraction and is any continuous function, then the composition (where is inclusion) has a fixed point. That is, there is such that . Since we have that and therefore

A topological space has the fixed-point property if and only if its identity map is universal.

A product of spaces with the fixed-point property in general fails to have the fixed-point property even if one of the spaces is the closed real interval.

The FPP is a topological invariant, i.e. is preserved by any homeomorphism. The FPP is also preserved by any retraction.

According to Brouwer fixed point theorem every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.[1]

## References

- Kinoshita, S. On Some Contractible Continua without Fixed Point Property.
*Fund. Math.***40**(1953), 96–98

- Samuel Eilenberg, Norman Steenrod (1952).
*Foundations of Algebraic Topology*. Princeton University Press. - Schröder, Bernd (2002).
*Ordered Sets*. Birkhäuser Boston.