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In the mathematical field of
descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to othe ...
, 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 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 ...
. In practice, a pointclass is usually characterized by some sort of ''definability property''; for example, the collection of all
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s 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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
and
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conve ...
. Strong set-theoretic principles may be stated in terms of the
determinacy Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and simil ...
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 A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such th ...
, and the
perfect set property In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a per ...
.


Basic framework

In practice, descriptive set theorists often simplify matters by working in a fixed Polish space such as
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 ...
or sometimes
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 ...
, each of which has the advantage of being zero dimensional, and indeed
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
to its finite or countable powers, so that considerations of dimensionality never arise.
Yiannis Moschovakis Yiannis Nicholas Moschovakis ( el, Γιάννης Μοσχοβάκης; born January 18, 1938) is a set theorist, descriptive set theorist, and recursion (computability) theorist, at UCLA. His book ''Descriptive Set Theory'' (North-Holland) is ...
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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
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 Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, 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 In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called ...
, and in the more complex
projective hierarchy In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is * \boldsymbol^1_1 if A is analytic * \boldsymbol^1_n if the complement of A, X\set ...
, are represented by sub- and super-scripted Greek letters in
boldface In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them. It is the equivalent of prosody stress in speech. Methods and use The most common methods in W ...
fonts; for example, \boldsymbol^0_1 is the pointclass of all
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s, \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 set In the mathematical field of descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student . Definition There are several equivalent d ...
s. Sets in such pointclasses need be "definable" only up to a point. For example, every
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, th ...
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 An oracle is a person or agency considered to provide wise and insightful counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. As such, it is a form of divination. Description The word ''o ...
. 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 In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wa ...
; that is, given a set in the pointclass, its
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...
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 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 descriptive ...
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 for each 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 lightface \Sigma^0_1, also called ''effectively open'', if there is a computable set ''S'' of finite sequences of naturals such that the given set is the union of the sets 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 ordinal In mathematics, specifically computability and set theory, an ordinal \alpha is said to be computable or recursive if there is a computable well-ordering of a subset of the natural numbers having the order type \alpha. It is easy to check that ...
s. This produces that
hyperarithmetic hierarchy In recursion theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an import ...
, which is the lightface analog of the Borel hierarchy. (The finite levels of the
hyperarithmetic hierarchy In recursion theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an import ...
are known as the
arithmetical hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...
.) A similar treatment can be applied to the projective hierarchy. Its lightface analog is known as the
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 ...
.


Summary

Each class is at least as large as the classes above it.


References

* {{cite book , author=Moschovakis, Yiannis N. , title=Descriptive Set Theory , url=https://archive.org/details/descriptivesetth0000mosc , url-access=registration , publisher=North Holland , year=1980 , isbn=0-444-70199-0 Descriptive set theory General topology