Fixed point property

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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.

1 Properties
2 Examples
- 2.1 The closed interval
- 2.2 The closed disc
3 References

[edit] 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.

[edit] Examples

[edit] The closed interval

The closed interval [0,1] has the fixed point property: Let f:[0,1] → [0,1] 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) = x² has no fixed point on the interval (0,1).

[edit] 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.

[edit] References

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