In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if
R
is a subset of
X
×
Y
, where
X
and
Y
are Polish spaces, then there is a subset
f
of
R
that is a partial function from
X
to
Y
, and whose domain (in the sense of the set of all
x
such that
f
(
x
)
exists) equals
{
x
∈
X
|
∃
y
∈
Y
(
x
,
y
)
∈
R
}
Such a function is called a uniformizing function for
R
, or a uniformization of
R
.
To see the relationship with the axiom of choice, observe that
R
can be thought of as associating, to each element of
X
, a subset of
Y
. A uniformization of
R
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
Γ
is said to have the uniformization property if every relation
R
in
Γ
can be uniformized by a partial function in
Γ
. The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.
It follows from ZFC alone that
Π
1
1
and
Σ
2
1
have the uniformization property. It follows from the existence of sufficient large cardinals that
Π
2
n
+
1
1
and
Σ
2
n
+
2
1
have the uniformization property for every natural number
n
.
Therefore, the collection of projective sets has the uniformization property.
Every relation in L(R) 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.)
