A
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
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 space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s on which every
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
mapping has a fixed point. But another use is in
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, where a
partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
''P'' is said to have the fixed point property if every
increasing function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
on ''P'' has a fixed point.
Definition
Let ''A'' be an object in the
concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects of the category as sets with additional ...
C. Then ''A'' has the ''fixed-point property'' if every
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
(i.e., every
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
)
has a fixed point.
The most common usage is when C = Top is the
category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...
. Then a topological space ''X'' has the fixed-point property if every continuous map
has a fixed point.
Examples
Singletons
In the
category of sets
In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
, the objects with the fixed-point property are precisely the
singletons.
The closed interval
The
closed interval
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
,1has the fixed point property: Let ''f'':
,1→
,1be 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
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two imp ...
, 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
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
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 the
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, the ...
.
Topology
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
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
, 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
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
its
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
is
universal
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company that is a subsidiary of Comcast
** Universal Animation Studios, an American Animation studio, and a subsidiary of N ...
.
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.
The FPP is a
topological invariant
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
, i.e. is preserved by any
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
. The FPP is also preserved by any
retraction.
According to the
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, the ...
, every
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
and
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
has the FPP. More generally, according to the
Schauder-Tychonoff fixed point theorem every
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
and
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
subset of a
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932
Borsuk Borsuk (the word for "badger" in a number of Slavic languages) may refer to:
*Angela Borsuk (born 1967), Israeli chess player
*Karol Borsuk, Polish mathematician
*Pavlo Borsuk, Ukrainian canoeist
*Borsuk, Hrubieszów County in Lublin Voivodeship (e ...
asked whether compactness together with
contractibility could be a sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.
[Kinoshita, S. On Some Contractible Continua without Fixed Point Property. ''Fund. Math.'' 40 (1953), 96–98]
References
*
*{{cite book , first = Bernd , last = Schröder , title = Ordered Sets , publisher = Birkhäuser Boston , year = 2002
Fixed points (mathematics)