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A mathematical object X has the fixed point property if every suitably well-behaved mapping from X to itself has a fixed point. It is a special case of the fixed morphism property. 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.

Properties

A retract of a space with the fixed point property also has the fixed point property.

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.

Examples

The closed interval

The closed interval has the fixed point property: Let f: → be a 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) = x has no fixed point on the interval (0,1).

The closed disc

The closed interval is a special case of the closed disc, which in any dimension has the fixed point property by the Brouwer fixed point theorem.

References

• Samuel Eilenberg, Norman Steenrod (1952). Foundations of Algebraic Topology. Princeton University Press.
• Schröder, Bernd (2002). Ordered Sets. Birkhäuser Boston.

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