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 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 has the fixed point property: Let f: → be a continuous mapping. If f = 0 or f = 1, then our mapping has a fixed point at 0 or 1. If not, then f > 0 and f − 1 < 0. Thus the function g = f − 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 x0 with g = 0, which is to say that f − x0 = 0, and so x0 is a fixed point.
The open interval does not have the fixed-point property. The mapping f = x2 has no fixed point on the interval.

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 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. 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.