Projective hierarchy

The projective hierarchy is analyzed in a mathematical branch of descriptive set theory; it is a for a specific education law gradually constructed hierarchy of sets whose lowest level begins with the Borel sets. Although the initial interest was the investigation of the subsets of the continuum, ie, the set of real numbers, but it has been shown that the theory can be just as easy to develop for Polish spaces, in particular can be then the Baire space presented in the below use education law. The projective hierarchy was introduced in 1925 by Lusin and Sierpiński.

Definition

The now following recursive definition follows structurally to the Borel hierarchy, to distinguish a one is here used as a superscript. stand for a Polish space, is the Baire space, ie the times Cartesian product of, provided with the product topology, where the usual in set theory term for the set of natural numbers is.

• , The class of all analytic sets.
• For, that is, the amounts are the complements of the sets.
• Is the class of all sets of the form, wherein the projection is made to the first component and a quantity.

A lot of that in one of the or the equivalent or lies, is called projective quantity.

Comments

• The above definition is recursive. You start with a class of analytic quantities, as explained by the class of complements and it means the specified projection technology. This is again explained as a class of complements, from which then results again using the above projection technology, and so on. The projection method for the definition of eponym for the projective hierarchy. is declared once and are explained.

• The use of the Baire space can in principle be avoided because it is homeomorphic to the space of irrational numbers that can be provided with a complete metric. The proof for the complete Metrisierbarkeit of irrational numbers is essentially the proof that quantities are Polish spaces again in Polish spaces. The metric to be constructed is not the Euclidean metric; hence the use of Baire space is more natural.

Properties

The quantities in the definition are complements of analytic sets, they are therefore also called koanalytisch. The amounts are of analytic sets, their complements are also analytical. By a theorem of Suslin, these are exactly the Borel sets.

The classes defined above satisfy the following inclusions

On an uncountable Polish space all specified inclusions are real. For most countable Polish space, however, all amounts are equal to the power set of the room.

All classes are closed under countable intersections and countable unions, in particular, is a - algebra. The projective sets as a whole, however, do not form a - algebra for uncountable Polish spaces. The projective hierarchy can, however, analogous to the Borel hierarchy to a ( rarer than the projective hierarchy under consideration ) hierarchy of countable for any ordinal continue. The union of all these quantities is the algebra of projective sets.

Is a Borel function between Polish spaces and belongs to one of the classes or so also.

Each amount in is Lebesgue measurable and has the Baire any amount property. Since transfer these properties to complements, this also applies to quantities. Further, any uncountable amount has a perfect subset, and therefore the cardinality of the continuum. For higher levels of the projective hierarchy, these properties are not provable in Zermelo -Fraenkel set theory with the axiom of choice. Gödel had shown that there are assuming the Konstruierbarkeitsaxioms a lot in that is not Lebesgue measurable, and an uncountable amount that contains no perfect subset. Further statements partly require stronger axioms, which go beyond the Zermelo -Fraenkel set theory, as described in sections 25 ( Descriptive Set Theory ) and 32 (More Descriptive Set Theory ) of the below textbook by Thomas Jech is executed. These properties are closely related to the determinism of certain games. In fact, it can be concluded for Borel sets from the Borel - determinacy, which is true in ZFC. Taking in addition to ZF the determinacy Axiom, whose relative consistency with ZF but not in ZFC is provable and that is contrary to the axiom of choice, so even all subsets of the real numbers are Lebesgue measurable, contain a non-empty perfect subset and have the Baire property. The demand of these properties for the class of projective sets, however, together with the axiom of choice possible by the determinacy of each game, the winning amount is a projective subset of the Baire space calls ( Axiom of projective determinacy ). This in turn follows from certain axioms about the existence of large cardinals. Already the determinacy of each game with analytical profit amount, however, can not prove in ZFC.