# Descriptive set theory

The branch of set theory whose subject is the study of sets in dependence of those operations by which these sets may be constructed from relatively simple sets (e.g. closed or open subsets of a given Euclidean, metric or topological space). These operations include union, intersection, taking a complement or projection, etc. Descriptive set theory was created in the early 20th century by the studies of E. Borel, R. Baire and H. Lebesgue in connection with the measurability of sets. Borel-measurable sets received the name of Borel sets or $B$- sets (cf. Borel set). On the other hand, Baire proposed a classification of functions, in so-called Baire function classes, and proved a number of theorems concerning these functions (cf. Baire classes; Baire theorem). Lebesgue showed that $B$- sets are identical to Lebesgue sets of Baire functions (cf. Lebesgue set), gave the first classification of $B$- sets and showed that none of its classes was empty.

The study of $B$- sets became an important task of descriptive set theory, and the first such problem was the cardinality of $B$- sets. After the introduction of the Lebesgue measure it was found that the class of measurable sets is much wider than the class of $B$- sets, and there arose the problem of determining whether or not a given set is measurable. The solution of this problem for a particular set usually involves the classification of the process by which this set may be constructed, i.e. its descriptive structure. This defines another important circle of problems dealt with by descriptive set theory, viz., to find the widest possible class (with preservation of measurability) of operations over sets and the study of the properties of the results of these operations. The solution of these problems, which arose as a results of studies by French mathematicians, must be credited, in essence, to Russian mathematicians, viz. N.N. Luzin and his school.

One of the most important problems, viz. the problem of the cardinality of $B$- sets, was solved by P.S. Aleksandrov  in 1916, who constructed the ${\mathcal A}$- operation for this purpose. He showed that the use of the ${\mathcal A}$- operation, while taking intervals as the starting point, makes it possible to construct any desired $B$- set, and that any uncountable set obtained by way of the ${\mathcal A}$- operation (and called an ${\mathcal A}$- set) contains a perfect set and thus has the cardinality of the continuum. This result was also independently obtained by F. Hausdorff. M.Ya. Suslin  showed that there exists an ${\mathcal A}$- set which is not a Borel set. He also introduced the name ${\mathcal A}$- set and ${\mathcal A}$- operation, in honour of Aleksandrov. ${\mathcal A}$- sets are also known as Suslin sets, or as analytic sets (cf. Analytic set). For an ${\mathcal A}$- set to be a $B$- set it is necessary and sufficient: 1) for its complement to be again an ${\mathcal A}$- set (the Suslin criterion); or 2) for it to be the result of an ${\mathcal A}$- operation with non-intersecting components (the Luzin criterion). All ${\mathcal A}$- sets are measurable and display the Baire property. The following new methods for obtaining ${\mathcal A}$- sets, equivalent to ${\mathcal A}$- operations, were found: ${\mathcal A}$- sets are projections of $B$- sets (and even of $G _ \delta$- sets); ${\mathcal A}$- sets are continuous images of the space $\mathbf I$ of irrational numbers; and, as a consequence, ${\mathcal A}$- sets are continuous images of $B$- sets . At the same time, a continuous one-to-one (and even a countably-to-multiple ) image of a $B$- set is a $B$- set, and any uncountable $B$- set is the union of an at most countable set and a one-to-one continuous image of $\mathbf I$. Finally, Luzin found yet another important way of defining ${\mathcal A}$- sets with the aid of his sieve operation (cf. Luzin sieve). Transfinite sieve indices and constituents have become a powerful tool in the study of the properties of ${\mathcal A}$- sets and their complements: $C {\mathcal A}$- sets (cf. $C {\mathcal A}$- set).

In the course of his studies on the cardinality of $C {\mathcal A}$- sets, Luzin introduced projective sets (cf. Projective set). Each class $\alpha$ of projective sets contains sets which do not belong to classes $< \alpha$, . The concept of a universal set , ,  is an important tool in the demonstration of this and other theorems asserting that certain classes of sets are non-empty. The study of projective sets, even those of the second class, encounters difficulties which have not yet been overcome. One problem which has thus remained unsolved is the measurability of $( B _ {2} )$- sets, the cardinality of these sets, and whether or not they display the Baire property. Important results in this matter were obtained by P.S. Novikov : There exists an uncountable $C {\mathcal A}$- set for which the assumption that it contains no perfect subset is not self-contradictory; there exists a $( B _ {2} )$- set for which the assumption that it is non-measurable is not self-contradictory.

The introduction by Aleksandrov  of the $\Gamma$- operation, which is complementary to the ${\mathcal A}$- operation, was the first step in the development of A.N. Kolmogorov's  and Hausdorff's  general theory of set-theoretic operations; however, the fundamental class of operations is constituted by positive set-theoretic operations, or the so-called $\delta$- $\sigma$- operations. For each such $\delta$- $\sigma$- operation $\Phi$ a complementary $\delta$- $\sigma$- operation $\Phi ^ {c}$ was defined and the formula

$$\Phi ^ {c} ( \{ E _ {n} \} ) = \ C [ \Phi ( \{ C E _ {n} \} ) ] ,$$

which may be considered as the definition of $\Phi ^ {c}$, was given. The concept of a normal $\delta$- $\sigma$- operation (for any family $M$ of sets $\Phi ( \Phi ( M) ) = \Phi ( M)$) was also introduced. The ${\mathcal A}$- operation and the $\Gamma$- operation are mutually complementary normal $\delta$- $\sigma$- operations. The same applies to the operations of countable union and countable intersection. One of the key theorems of the general theory of operations over sets in topological spaces is Kolmogorov's complements theorem: If a space contains a discontinuum, if $M$ is the system of all closed subsets of this space and if $\Phi$ is an arbitrary set-theoretic operation, then the complement of at least one set of the family $\Phi ( M)$ does not belong to it , , . The application to a given family $M$ of sets of $\delta$- $\sigma$- operations $\Phi$ and $\Phi ^ {c}$ in alternation makes it possible to construct an increasing transfinite sequence of classes $M _ {0} = M , M _ {1} \dots M _ \alpha \dots$ $\alpha < \omega _ {1}$, which satisfies the Kolmogorov theorem on non-emptyness of classes: If $\Phi$ is a normal $\delta$- $\sigma$- operation of greater cardinality than the operation of countable union (or of countable intersection), while $M$ is the family of all closed subsets of a metric space containing a discontinuum, then all the classes $M _ \alpha$, $\alpha < \omega _ {1}$, generated by the operation $\Phi$ from the family $M$ are pairwise different. Here, a $\delta$- $\sigma$- operation $\Phi$ is considered to be of greater cardinality than a $\delta$- $\sigma$- operation $\Psi$ if $\Phi ( M) \supseteq \Psi ( M)$ for any family $M$ of sets , . If $\Phi = \cup _ {1} ^ \infty$( or $\cap _ {1} ^ \infty$), then the classes $M _ \alpha$, $\alpha < \omega _ {1}$, generated by the operation $\Phi$ on the family $M$ represent the classes of $B$- sets generated by the family $M$( the Hausdorff classification). In a similar manner, an ${\mathcal A}$- operation generates classes $M _ \alpha$, $\alpha < \omega _ {1}$, of $C$- sets (Luzin sets, cf. Luzin set) out of the family $M$ of Borel sets. All $C$- sets are measurable and have the Baire property. All $C$- sets form part of the class $( B _ {2} )$, but do not exhaust it.

The concept of separation , introduced by Luzin, plays a highly important part in descriptive set theory. The first separation principle: Any two non-intersecting ${\mathcal A}$- sets are $( B)$ separable. The second separation principle: If $E$ and $P$ are two ${\mathcal A}$- sets (or $C {\mathcal A}$- sets), then the sets $E \setminus P$ and $P \setminus E$ are ( $C {\mathcal A}$) separable. There exists two $C {\mathcal A}$- sets which are not $( B)$ separable. The problem of separation of second-class projective sets was solved by Novikov  and its separation principles are invertible: they are obtained from the respective theorems for projective sets of the first class by replacing $( {\mathcal A} )$ by $( C {\mathcal A} _ {2} )$, $( C {\mathcal A} )$ by $( {\mathcal A} _ {2} )$ and $( B)$ by $( B _ {2} )$. Novikov  also solved the separation problem in the class of $C$- sets (see also ). He also generalized the concept of separation of sets to include the concept of simultaneous or multiple separation of sets, as well as the principle of equality of indices , which is a principal instrument in proving the theorems on multiple separation.

An important stage in the development of descriptive set theory was the solution of the uniformization problem. A set $P$ uniformizes a set $E \subset X \times Y$ if $P \subset E$, if $P$ has the same projection on $X$ as does $E$ and if it is projected on $X$ in a one-to-one manner. All $B$- sets are uniformized by $C {\mathcal A}$- sets. The process of effective selection of a point  in a non-empty $C {\mathcal A}$- set yielded a stronger result: All $C {\mathcal A}$- sets are uniformized by a $C {\mathcal A}$- set. Any ${\mathcal A}$- set is uniformized by a set of type ${\mathcal A} _ {\rho \sigma \theta }$, . There exists an ${\mathcal A}$- set in the Euclidean plane which is not uniformized either by an ${\mathcal A}$- set or by a $C {\mathcal A}$- set (see, for example, ). The problem of the conditions under which a $B$- set can be uniformized by a $B$- set can be most generally answered as follows: Any plane $B$- set intersecting the straight lines $x = {\textrm{ const } }$ in $F _ \sigma$- sets is uniformized by a $B$- set. The problem of uniformization arose in solving the problem of implicit $B$- functions . Other problems arose at the same time: the nature of projections of $B$- sets, splitting of sets, covering of sets, and the nature of the set of all projection points of a given $B$- set $E$ whose pre-images (in intersection with $E$) display a certain given property. The following theorems  give some idea of the second and third problems: a $B$- set with a countable-to-multiple projection is the union of a countable number of uniform $B$- sets and, for any two such sets, one of them will lie beneath the other (splitting theorem of a $B$- set); any ${\mathcal A}$- set with a countable-to-multiple projection is contained in some $B$- set with this property (covering theorem of an ${\mathcal A}$- set).

In addition to the Lebesgue and Hausdorff classification of $B$- sets, there is also their classification according to Luzin–de la Vallée Poussin. In this classification, the structure of sets of a given class $K _ \alpha$, $\alpha < \omega _ {1}$, is studied by means of sets which can be represented as an intersection but not as a union of a countable number of sets of classes $< \alpha$; these sets are said to be elements of class $\alpha$. Any set of class $K _ \alpha$ is the union of a countable number of pairwise non-intersecting elements of classes $\leq \alpha$. Each class $K _ \alpha$ can be subdivided into subclasses $K _ \alpha ^ \beta$, $\beta < \omega _ {1}$, each class containing subclasses with numbers $\beta < \omega _ {1}$ which may be arbitrarily large. Since each set of class $\alpha$ consists of elements, the problem arose of studying the elements themselves; in particular, of the presence in each class $K _ \alpha$ of a basic topological type of elements, to be called canonical, such that each element of class $\alpha$ may be represented as the union of a countable number of canonical elements of classes $\leq \alpha$( everything being considered in the space $\mathbf I$ of irrational numbers). The first class includes canonical elements of two types: a one-point set and a topological image of a perfect Cantor set. Each element in the second class  is the union of a canonical element of class 2, which is a set homeomorphic to the space $\mathbf I$, and a set of class $\leq 1$. Canonical elements of the third class have also been found; these canonical elements have been constructed by Baire . The problem of existence of canonical elements of higher classes, which was very difficult, was solved by L.V. Keldysh , who proved that canonical elements exist in each class $\alpha < \omega _ {1}$ and clarified their structure. Each element of class $\alpha > 2$ is the union of one canonical element of class $\alpha$ with at most a countable number of sets of classes $< \alpha$. Yet another difficult problem in this field is connected with the construction of arithmetical examples of $B$- sets of lower classes. Baire  gave such an example for the class 3. Keldysh  gave arithmetical examples of elements of all finite classes (see also ), and pointed to the possibility, in principle, of constructing such examples for classes $\alpha \geq \omega _ {0}$.

An important part in descriptive set theory is played by the Lavrent'ev theorem on the extension of homeomorphisms : Let $X$ and $Y$ be complete metric spaces, let $A \subset X$, $B \subset Y$ and let $f : A \rightarrow B$ be a homeomorphism; then there exists an extension of $f$ to a homeomorphism of two $G _ \delta$- subsets of these spaces. This theorem readily leads to the theorem of topological invariance : Let $\mathfrak B$ be a system of closed sets and let $\Phi$ be a $\delta$- $\sigma$- operation such that intersection of a $\Phi ( \mathfrak B )$- set (i.e. of a set of the family $\Phi ( \mathfrak B )$) with a $G _ \delta$- set is again a $\Phi ( \mathfrak B )$- set; then each $\Phi ( \mathfrak B )$- subset of a complete metric space is a $\Phi ( \mathfrak B )$- set in any metric space in which it is topologically contained. In this theorem, $\mathfrak B$ may be replaced by a system $\mathfrak A$ of open sets. Thus, $\Phi ( \mathfrak B )$- sets for which $\Phi$ satisfies the above condition and which form part of complete metric spaces are absolute $\Phi ( \mathfrak B )$- sets, and the same applies to $\Phi ( \mathfrak A )$- sets (as regards the absolute nature in the class of metric spaces). In a number of special cases, such as $C {\mathcal A}$- sets  ( $\Phi$ is a $\Gamma$- operation), this result has been established without having recourse to Lavrent'ev's theorem. Any complete metric space is an absolute $G _ \delta$. Any metrizable absolute $G _ \delta$ is homeomorphic to some complete metric space (the Aleksandrov–Hausdorff theorem ).

The following Aleksandrov–Urysohn theorem on the topological characterization of the space of irrational numbers is valid: Each zero-dimensional metric space $X$ with a countable base which has no points of local compactness and which is an absolute $G _ \delta$, is homeomorphic to the space $\mathbf I$. Since $\mathbf I$ is an absolute $G _ \delta$ and has all the other properties of the space $X$, it follows that the properties of $X$ listed in the theorem represent a full topological characterization of $\mathbf I$. This characterization led to the following result : Any metric space which is a continuous image of $\mathbf I$( and thus, an absolute ${\mathcal A}$- set) is also a quotient image of $\mathbf I$. One may also quote the following related results: A non-empty separable metric space is a continuous open image of $\mathbf I$ if and only if it is an absolute $G _ \delta$; here, "open" may be replaced by "closed" . The concept of a $B$- measurable mapping or a $B$- function, in particular, of a $B$- measurable mapping of class $\alpha$, is a generalization of a continuous mapping; the concepts of a generalized homeomorphism of class $( \alpha , \beta )$ and of a $B$- isomorphism are generalizations of a homeomorphism. For such mappings see .

In formulating the results, the class of spaces to which they are applicable is usually not specified. This is explained by the fact that most of the classical results were obtained for subsets of $\mathbf I$, but almost-all of them (save specified exceptions) remain valid if $\mathbf I$ is replaced by any separable absolute $G _ \delta$( in particular, by a complete metric space with a countable base). Further developments in descriptive set theory involved the generalization of the classical results to include the following cases: 1) complete metric spaces (not necessarily separable); 2) perfectly-normal topological spaces, including, in particular, compact Hausdorff spaces; and 3) general topological spaces. Even in the first case the generalization of the classical theory encounters serious difficulties, and is frequently altogether impossible. A.H. Stone  considered the generalization of the theory of $B$- sets and ${\mathcal A}$- sets to include this case.

The class of perfectly-normal spaces is a class outside which the important fact that the system of $B$- sets ( ${\mathcal A}$- sets) generated by the closed sets of the given space coincides with the system of $B$- sets ( ${\mathcal A}$- sets) generated by the open sets of this space, is no longer true. The following important theorem on non-emptiness of classes  is valid: In any uncountable perfectly-normal compact Hausdorff space there exists for each class $\alpha < \omega _ {1}$ a $B$- set of class $\alpha$ which is not a $B$- set of class $< \alpha$.

The problem of uniformization has become a partial case of the general problem on the section of a multi-valued mapping (cf. Section of a mapping).

The modern development of descriptive set theory in general topological spaces (third case) is connected with the needs of other domains of mathematics (e.g. potential theory). For more details see  (in which there is an extensive bibliography).

The ideas and methods of descriptive set theory had a profound effect on the development of several domains in mathematics: analysis, the theory of functions, topology, mathematical logic, etc.

How to Cite This Entry:
Descriptive set theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Descriptive_set_theory&oldid=51316
This article was adapted from an original article by A.G. El'kinV.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article