- 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. It is a special case of the
fixed morphism property . The term is most commonly used to describetopological space s on which every continuous mapping has a fixed point. But another use is inorder theory , where apartially ordered set "P" is said to have the fixed point property if everyincreasing function on "P" has a fixed point.Definition
Let "A" be an object in the
concrete category C. Then "A" has the "fixed point property" if everymorphism (i.e., every function) 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.
Properties
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.
Examples
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 theintermediate value theorem , there is some point "x"0 with "g"("x"0) = 0, which is to say that "f"("x"0) − "x"0 = 0, and so "x"0 is a fixed point.The
open interval does "not" have the fixed point property. The mapping "f"("x") = "x"2 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 finite dimension has the fixed point property by theBrouwer fixed point theorem .References
*cite book | first = Norman Steenrod | last = Samuel Eilenberg | title = Foundations of Algebraic Topology | publisher = Princeton University Press | year = 1952
*cite book | first = Bernd | last = Schröder | title = Ordered Sets | publisher = Birkhäuser Boston | year = 2002
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