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. Let A be an object in the 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 . Then a topological space X has the fixed-point property if every continuous map has a fixed point. In the , the objects with the fixed-point property are precisely the singletons. 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). 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. 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.

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