A mathematical has the fixed-point property if every suitably well-behaved mapping 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 is said to have the fixed point property if every increasing function has a fixed point.

Definition

be an object in the concrete category has the fixed-point property if every morphism (i.e., every function has a fixed point.

is the category of topological spaces 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. By the intermediate value theorem is a fixed point.

The open interval does not.

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 is any continuous function

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.

References