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. 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 concrete category C. Then A has the fixed-point property if every morphism (i.e., every function) f : A → A {displaystyle f:A o A} 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 f : X → X {displaystyle f:X o X} has a fixed point. In the category of sets, the objects with the fixed-point property are precisely the singletons. The closed interval has the fixed point property: Let f: → 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 r : X → A {displaystyle r:X o A} is a retraction and f : A → A {displaystyle f:A o A} is any continuous function, then the composition i ∘ f ∘ r : X → X {displaystyle icirc fcirc r:X o X} (where i : A → X {displaystyle i:A o X} is inclusion) has a fixed point. That is, there is x ∈ A {displaystyle xin A} such that f ∘ r ( x ) = x {displaystyle fcirc r(x)=x} . Since x ∈ A {displaystyle xin A} we have that r ( x ) = x {displaystyle r(x)=x} and therefore f ( x ) = x . {displaystyle f(x)=x.} A topological space has the fixed-point property if and only if its identity map is universal.