In the mathematical field of descriptive set theory, a subset
A
of a Polish space
X
is projective if it is
Σ
n
1
for some positive integer
n
. Here
A
is
Σ
1
1
if
A
is analytic
Π
n
1
if the complement of
A
,
X
∖
A
, is
Σ
n
1
Σ
n
+
1
1
if there is a Polish space
Y
and a
Π
n
1
subset
C
⊆
X
×
Y
such that
A
is the projection of
C
; that is,
A
=
{
x
∈
X
∣
∃
y
∈
Y
(
x
,
y
)
∈
C
}
The choice of the Polish space
Y
in the third clause above is not very important; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.
There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters
Σ
and
Π
) and the projective hierarchy on subsets of Baire space (denoted by boldface letters
Σ
and
Π
). Not every
Σ
n
1
subset of Baire space is
Σ
n
1
. It is true, however, that if a subset X of Baire space is
Σ
n
1
then there is a set of natural numbers A such that X is
Σ
n
1
,
A
. A similar statement holds for
Π
n
1
sets. Thus the sets classified by the projective hierarchy are exactly the sets classified by the relativized version of the analytical hierarchy. This relationship is important in effective descriptive set theory.
A similar relationship between the projective hierarchy and the relativized analytical hierarchy holds for subsets of Cantor space and, more generally, subsets of any effective Polish space.
