Fixed-point property

A mathematical object ${\displaystyle X}$ has the fixed-point property if every suitably well-behaved mapping from ${\displaystyle X}$ 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 ${\displaystyle P}$ is said to have the fixed point property if every increasing function on ${\displaystyle P}$ has a fixed point.

Definition

Let ${\displaystyle A}$ be an object in the concrete category ${\displaystyle \mathbf {C} }$. Then ${\displaystyle A}$ has the fixed-point property if every morphism (i.e., every function) ${\displaystyle f:A\to A}$ has a fixed point.

The most common usage is when ${\displaystyle \mathbf {C} =\mathbf {Top} }$ is the category of topological spaces. Then a topological space ${\displaystyle X}$ has the fixed-point property if every continuous map ${\displaystyle f:X\to X}$ 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 ${\displaystyle [0,1]}$ has the fixed point property: Let ${\displaystyle f:[0,1]\to [0,1]}$ be a continuous mapping. If ${\displaystyle f(0)=0}$ or ${\displaystyle f(1)=1}$, then our mapping has a fixed point at 0 or 1. If not, then ${\displaystyle f(0)>0}$ and ${\displaystyle f(1)-1<0}$. Thus the function ${\displaystyle f(g)=f(x)-x}$ is a continuous real valued function which is positive at ${\displaystyle x=0}$ and negative at ${\displaystyle x=1}$. By the intermediate value theorem, there is some point ${\displaystyle x_{0}}$ with ${\displaystyle g(x_{0})=0}$, which is to say that ${\displaystyle f(x)-x=0}$, and so ${\displaystyle x_{0}}$ is a fixed point.

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

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 ${\displaystyle A}$ of a space ${\displaystyle X}$ with the fixed-point property also has the fixed-point property. This is because if ${\displaystyle r:X\to A}$ is a retraction and ${\displaystyle f:A\to A}$ is any continuous function, then the composition ${\displaystyle i\circ f\circ r:X\to X}$ (where ${\displaystyle i:A\to X}$ is inclusion) has a fixed point. That is, there is ${\displaystyle x\in A}$ such that ${\displaystyle f\circ r(x)=x}$. Since ${\displaystyle x\in A}$ we have that ${\displaystyle r(x)=x}$ and therefore ${\displaystyle f(x)=x.}$

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 the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector 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]