Descriptive set theory

The descriptive set theory is a branch of set theory, which is concerned with properties of definable sets. Here first are subsets of real numbers as sets, Borel sets and derived quantity hierarchies in the foreground; the set-theoretic, topological or measure theoretic properties can be studied in general polish spaces just as well, with the homeomorphic to the set of irrational numbers Baire space plays a special role. The basic idea is, starting by certain education laws to construct of "simple" complex quantities and quantities to study their properties. The occurring in mathematical practice quantities can be obtained in this way.

Historical beginnings

An important question of set theory from the beginning was the problem of the cardinality of the continuum, ie, the set of real numbers. The continuum hypothesis that there are no other widths between the cardinality of countably infinite sets and the cardinality of the continuum, has been found by the work of Gödel and Cohen as neither provable nor refutable. This obviously does not mean that you can show for certain types of subsets of the continuum that they automatically have the cardinality of the continuum in the uncountable case; we say then that this type of sets satisfies the continuum hypothesis. Especially easy is that for open sets in, because these are associations of open intervals. An open set is therefore either empty or contains an open interval and is thus equal to powerful; ie the open sets satisfy the continuum hypothesis. For closed sets, ie for the complements of open sets, which is a little more difficult. A very early result in this direction is defined by the set of Cantor - Bendixson, in fact satisfy the amounts of the continuum hypothesis concluded.

Baire had in 1899 introduced what we now call Baire functions; This is the smallest set of functions or other polish rooms containing all continuous functions and is closed under pointwise convergence. Lebuesge characterized the 1905 so-called analytically representable, that is, the smallest set of functions containing all constants and all projections and is closed under sums, products and pointwise convergence. In this context, he introduced the Borel sets, claiming in a lemma that projections of Borel sets are those again. But that this is wrong, Suslin was noticed, which developed the concept of the analytic set. Even for analytic sets could be shown that they satisfy the continuum hypothesis. For larger classes, which means certain education laws gain from the analytical and can be arranged in so-called hierarchies, the question remains open.

The branch of effective descriptive set theory owes much to developments Stephen Cole Kleene, about the development of the arithmetical hierarchy, the connections to the classical descriptive set theory, however, were revealed only later.

Hierarchies

The following remarks are intended to give a first impression of the research field of descriptive set theory.

Borel hierarchy

Starting point of the Borel hierarchy is the class of open sets in general or in a perfect, Polish space; the class of open sets is denoted by. Is the set of natural numbers with the discrete topology, so is again a Polish space. will now be defined as the set of all projections of complements on the first component, that is, consisting of all the sets of the form, wherein an amount, that is an open volume, and the projection is on the first component. This process can be iterated by using, as the class of all sets of the form, said all subsets of passes, their complements, are amounts.

The complements of the form class of sets. The amounts are also known as quantities and their complements, that is, amounts as amounts. Overall, we obtain by means of the above mode of formation rising classes

And it can be shown that this construction is not led out to Borel quantities and that in addition

Applies. This raises the question of whether coincides with the class of all Borel sets. The answer is no, you have to continue the educational process transfinit above, which can be carried out informally with the concept of ordinal. It then turns out that you have to perform this process once, with the smallest uncountable ordinal (see also Aleph function) to obtain in this way all Borel sets.

Projective hierarchy

The projective hierarchy is created according to the same pattern from the class of open sets, only the space is replaced by the Baire space, the set of all functions is what you, as usual, identified themselves with the times Cartesian product of and on the product topology considered. This space is homeomorphic to the space of irrational numbers with the relative topology of why you are in the descriptive set theory is often called the space of irrational numbers the Baire space. Are the names of the hierarchies

Note that, the upper index is a one. So is the class of all sets of the form, where all closed subsets of a Polish space and passes is, these quantities are also called analytic. is again the class of complements of such sets, which are therefore also called koanalytisch.

Already Suslin showed that exactly matches the Borel sets. It can be shown that the quantities satisfy the continuum hypothesis and all are Lebesgue measurable. Those statements for lost; Gödel had shown that there are assuming the Konstruierbarkeitsaxioms a lot in that is not Lebesgue measurable. After a set of Sierpiński is any association of - many Borel sets.

κ - Suslin sets

If, in the construction of the Lusin - Baire space by the hierarchy, which is a cardinal with the discrete topology is, so to get to the concept of - Suslin set. A subset of a Polish space is a Suslin - set if it has the form of a closed set. The class of all such sets is denoted by.

Appears to be consistent with, ie with the class of all analytic quantities match. After a set of Shoenfield each is a - Suslin set. Statements about these set classes require deeper methods of set theory, there often arises the question of sufficiently strong axioms of set theory.

Regularity

In addition to such amounts arising from certain operations, we consider certain regularity properties of subsets of Polish spaces and their relation to the amounts provided through these structures. Examples of such properties are:

• A lot has the Baire property if it differs by only a meager amount of an open set.
• A set is universally measurable if it is measurable with respect to each complete, finite measure, which is defined for all Borel sets.
• A lot has the perfect amounts of property if it is countable or contains a non-empty perfect amount.

Further Questions

Of course, more important questions of descriptive set theory concerning the functions between Polish spaces, particularly their Messbarkeitseigenschaften, and equivalence relations and algebraic structures on Polish spaces. Further, the formation processes described above can be examined for their predictability out this happens in the recursion closely integrated subspecialty of effective descriptive set theory.

Applications

Common applications for the descriptive set theory about in the following areas:

• Operator algebras
• Ergodic Theory
• Theory of infinite automata and games