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.
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.
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.
Yiannis 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.
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
is the collection of open sets, not on the spaces themselves).
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,
is the collection of all sets that are simultaneously Fσ and Gδ, and
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
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.
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.
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 collection of sets of the form {x∈ωω
s is an initial segment of x} for each fixed finite sequence s of natural numbers), then the open, or
, sets may be characterized as all (arbitrary) unions of basic open neighborhoods.
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.
sets (that is, there is a computable enumeration of indices of
This relationship between lightface sets and their indices is used to extend the lightface Borel hierarchy into the transfinite, via recursive ordinals.
A similar treatment can be applied to the projective hierarchy.