In the mathematical field of
descriptive set theory, a subset
of a
Polish 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 be ...
is projective if it is
for some positive integer
. Here
is
*
if
is
analytic
*
if the
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of
,
, is
*
if there is a Polish space
and a
subset
such that
is the projection of
; that is,
The choice of the Polish space
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
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
or
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
or the
real line.
Relationship to the analytical hierarchy
There is a close relationship between the relativized
analytical hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers ...
on subsets of Baire space (denoted by lightface letters
and
) and the projective hierarchy on subsets of Baire space (denoted by boldface letters
and
). Not every
subset of Baire space is
. It is true, however, that if a subset ''X'' of Baire space is
then there is a set of natural numbers ''A'' such that ''X'' is
. A similar statement holds for
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
Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightface definitions; that is, definitions that do not require an arbitrary real parameter (Moschovakis 1980). Thus effective descriptiv ...
.
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 In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in effective descriptive set theory and in constructive analysis. In particular, standard examples ...
.
Table
References
*
* {{Citation , last1=Rogers , first1=Hartley , author-link= Hartley Rogers, title=The Theory of Recursive Functions and Effective Computability , orig-year=1967 , publisher=First MIT press paperback edition , isbn=978-0-262-68052-3 , year=1987
Descriptive set theory
Mathematical logic hierarchies