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.
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.
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 x0 with g(x0) = 0, which is to say that f(x0) − x0 = 0, and so x0 is a fixed point.
The open interval does not have the fixed-point property. The mapping f(x) = x2 has no fixed point on the interval (0,1).
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.
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.
- 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.