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Uniformization (set theory)
Encyclopedia
In set theory
, the axiom of uniformization, a weak form of the axiom of choice, states that if
is a subset
of
, where 
and 
are Polish space
s,
then there is a subset
of 
that is a partial function
from
to 
, and whose domain (in the sense of the set of all 
such that 
exists) equals
Such a function is called a uniformizing function for
, or a uniformization of 
.
To see the relationship with the axiom of choice, observe that 
can be thought of as associating, to each element of 
, a subset of 
. A uniformization of 
then picks exactly one element from each such subset, whenever the subset is nonempty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC.
A pointclass
is said to have the uniformization property if every relation 
in 
can be uniformized by a partial function in 
. The uniformization property is implied by the scale property
, at least for adequate pointclass
es of a certain form.
It follows from ZFC alone that
and 
have the uniformization property. It follows from the existence of sufficient large cardinals that
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the axiom of uniformization, a weak form of the axiom of choice, states that if
is a subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of
, where and are Polish spacePolish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
s,
then there is a subset
of that is a partial functionPartial function
In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
from
to , and whose domain (in the sense of the set of all such that exists) equals
Such a function is called a uniformizing function for
, or a uniformization of . can be thought of as associating, to each element of , a subset of . A uniformization of then picks exactly one element from each such subset, whenever the subset is nonempty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC.A pointclass
Pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by some sort of definability property; for example, the...
is said to have the uniformization property if every relation in can be uniformized by a partial function in . The uniformization property is implied by the scale propertyScale property
In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space...
, at least for adequate pointclass
Adequate pointclass
In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursive pointsets and is closed under recursive substitution, bounded universal and existential quantification and preimages by recursive functions....
es of a certain form.
It follows from ZFC alone that
and have the uniformization property. It follows from the existence of sufficient large cardinals that
and 
have the uniformization property for every natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
.- Therefore, the collection of projective sets has the uniformization property.
- Every relation in L(R)L(R)In set theory, L is the smallest transitive inner model of ZF containing all the ordinals and all the reals. It can be constructed in a manner analogous to the construction of L , by adding in all the reals at the start, and then iterating the definable powerset operation through all the...
can be uniformized, but not necessarily by a function in L(R). In fact, L(R) does not have the uniformization property (equivalently, L(R) does not satisfy the axiom of uniformization).- (Note: it's trivial that every relation in L(R) can be uniformized in V, assuming V satisfies AC. The point is that every such relation can be uniformized in some transitive inner model of V in which AD holds.)