# User Contributed Dictionary

# Extensive Definition

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 collection of
all open
sets in some fixed collection of Polish spaces is a pointclass.
(An open set may be seen as in some sense definable because it
cannot be a purely arbitrary collection of points; for any point in
the set, all points sufficiently close to that point must also be
in the set.)

Pointclasses find application in formulating many
important principles and theorems from set theory and
real
analysis. Strong set-theoretic principles may be stated in
terms of the determinacy of various
pointclasses, which in turn implies that sets in those pointclasses
(or sometimes larger ones) have regularity properties such as
Lebesgue
measurability (and indeed universal
measurability), the property
of Baire, and the perfect
set property.

## Basic framework

In practice, descriptive set theorists often simplify matters by working in a fixed Polish space such as Baire space or sometimes Cantor space, each of which has the advantage of being zero dimensional, and indeed homeomorphic to its finite or countable powers, so that considerations of dimensionality never arise. Moschovakis provides greater generality by fixing once and for all a collection of underlying Polish spaces, including the set of all naturals, the set of all reals, Baire space, and Cantor space, and otherwise allowing the reader to throw in any desired perfect Polish space. Then he defines a product space to be any finite Cartesian product of these underlying spaces. Then, for example, the pointclass \boldsymbol^0_1 of all open sets means the collection of all open subsets of one of these product spaces. This approach prevents \boldsymbol^0_1 from being a proper class, while avoiding excessive specificity as to the particular Polish spaces being considered (given that the focus is on the fact that \boldsymbol^0_1 is the collection of open sets, not on the spaces themselves).## Boldface pointclasses

The pointclasses in the Borel hierarchy, and in the more complex projective hierarchy, are represented by sub- and super-scripted Greek letters in boldface fonts; for example, \boldsymbol^0_1 is the pointclass of all closed sets, \boldsymbol^0_2 is the pointclass of all Fσ sets, \boldsymbol^0_2 is the collection of all sets that are simultaneously Fσ and Gδ, and \boldsymbol^1_1 is the pointclass of all analytic sets.Sets in such pointclasses need be "definable"
only up to a point. For example, every singleton
set in a Polish space is closed, and thus \boldsymbol^0_1.
Therefore it cannot be that every \boldsymbol^0_1 set must be "more
definable" than an arbitrary element of a Polish space (say, an
arbitrary real number, or an arbitrary countable sequence of
natural numbers). Boldface pointclasses, however, may (and in
practice ordinarily do) require that sets in the class be definable
relative to some real number, taken as an oracle. In
that sense, membership in a boldface pointclass is a definability
property, even though it is not absolute definability, but only
definability with respect to a possibly undefinable real
number.

Boldface pointclasses, or at least the ones
ordinarily considered, are closed under Wadge
reducibility; that is, given a set in the pointclass, its
inverse
image under a continuous
function (from a product space to the space of which the given
set is a subset) is also in the given pointclass. Thus a boldface
pointclass is a downward-closed union of Wadge
degrees.

## Lightface pointclasses

The Borel and projective hierarchies have analogs in effective descriptive set theory in which the definability property is no longer relativized to an oracle, but is made absolute. For example, if one fixes some collection of basic open neighborhoods (say, in Baire space, the set of all sets of the form for any fixed finite sequence s of natural numbers), then the open, or \boldsymbol^0_1, sets may be characterized as all (arbitrary) unions of basic open neighborhoods. The analogous \Sigma^0_1 sets, with a lightface \Sigma, are no longer arbitrary unions of such neighborhoods, but computable unions of them (that is, a set is \Sigma^0_1 if there is a computable set S of finite sequences of naturals such that the given set is the union of all for s in S). A set is lightface \Pi^0_1 if it is the complement of a \Sigma^0_1 set. Thus each \Sigma^0_1 set has at least one index, which describes the computable function enumerating the basic open sets from which it is composed; in fact it will have infinitely many such indices. Similarly, an index for a \Pi^0_1 set B describes the computable function enumerating the basic open sets in the complement of B.A set A is lightface \Sigma^0_2 if it is a union
of a computable sequence of \Pi^0_1 sets (that is, there is a
computable enumeration of indices of \Pi^0_1 sets such that A is
the union of these sets). This relationship between lightface sets
and their indices is used to extend the lightface Borel hierarchy
into the transfinite, via recursive
ordinals. This produces that hyperarithmetic
hierarchy, which is the lightface analog of the Borel
hierarchy. (The finite levels of the hyperarithmetic
hierarchy are known as the arithmetical
hierarchy.)

A similar treatment can be applied to the
projective hierarchy. Its lightface analog is known as the analytical
hierarchy.

## References

- Descriptive Set Theory