# Projective set

A set that can be obtained from Borel sets (cf. Borel set) by repeated application of the operations of projection and taking complements. Projective sets are classified in classes forming the projective hierarchy. Let $I = \omega ^ \omega$ be the Baire space (homeomorphic to the space of irrational numbers). A set $P \subset I ^ {m}$ belongs to: 1) the class $A _ {1}$ if $P$ is the projection of a Borel set in the space $I ^ {m+} 1$; 2) the class $CA _ {n}$( $P$ is a $CA _ {n}$- set) if its complement $I ^ {m} \setminus P$ is an $A _ {n}$- set $( n \geq 1 )$; 3) the class $A _ {n}$( $P$ is an $A _ {n}$- set) if $P$ is the projection of a $CA _ {n-} 1$- set in the space $I ^ {m+} 1$, $n \geq 2$; and 4) the class $B _ {n}$ if $P$ belongs simultaneously to the classes $A _ {n}$ and $CA _ {n}$, $n \geq 1$. The same classes are obtained when the projection is replaced by a continuous image (of a set in the same space $I ^ {m}$).

By virtue of the Suslin theorem, the class $A _ {1}$ coincides with the class of ${\mathcal A}$- sets (consequently, the class $C A _ {1}$ coincides with the class of $C {\mathcal A}$- sets, cf. ${\mathcal A}$- set; $C {\mathcal A}$- set), and the class $B _ {1}$ coincides with the class of Borel sets. For every class $A _ {n}$ a universal set is constructed and used to prove the following projective hierarchy theorem ( "existence" theorem, theorem on "non-emptiness of the classes" ): $B _ {n} \subset A _ {n} \subset B _ {n+} 1$( consequently, $A _ {n} \subset B _ {n+} 1 \subset A _ {n+} 1$), where each inclusion is strict. The cardinality of the set of all projective sets of the space $I$ equals $2 ^ {\aleph _ {0} }$.

Every $A _ {2}$- set is a union of $\aleph _ {1}$ Borel sets and hence is either countable, or has cardinality $\aleph _ {1}$ or $2 ^ {\aleph _ {0} }$( see  and  ). For the class $A _ {2}$ the uniformization and reduction principles hold, while for the class $CA _ {2}$ the (first) separation principle holds; cf. Descriptive set theory. Each projective class with index $n \geq 2$ is invariant under the ${\mathcal A}$- operation. For each of the classes $A _ {n}$, $CA _ {n}$ there is a $\delta$- $\sigma$- operation that yields exactly all the sets of the class, starting from closed sets. The study of projective sets (even of the second class) is a difficult problem. Many problems in the theory of projective sets turned out to be undecidable in the classical sense, which completely confirmed the foresight (see ): "The field of projective sets is a field where the principle of the excluded middle is not strong enough any more" . The theory of projective sets was advanced further by involving strong set-theoretic assumptions, such as $\mathop{\rm MC}$( there exists a measurable cardinal), $\mathop{\rm PD}$( the projective determinacy axiom) and $V = L$( the constructibility axiom, cf. Gödel constructive set).

Under the assumption $\mathop{\rm MC}$: Every $A _ {2}$- set is (Lebesgue-) measurable, has the Baire property and, if uncountable, contains a (non-empty) perfect subset; every $A _ {3}$- set can be uniformized by an $A _ {4}$- set.

Under the assumption $\mathop{\rm PD}$: a) Every projective set is measurable, has the Baire property and, if uncountable, contains a perfect subset and can be uniformized by a projective set; more precisely, the uniformization principle holds for the classes $A _ {2n}$ and $CA _ {2n+} 1$. b) For the classes $A _ {2n}$ and $CA _ {2n+} 1$ the reduction principle holds, hence for the classes $A _ {2n+} 1$ and $CA _ {2n}$ the separation principle holds.

Under the assumption $V = L$: a) There exists an uncountable $CA$- set that does not contain a perfect subset, and there exists a non-measurable $B _ {2}$- set without the Baire property. b) For $n \geq 2$ the uniformization principle holds for the class $A _ {n}$.

If for a class $A _ {n}$ the uniformization principle holds, then so does the reduction principle. For $n \geq 3$ the reverse implication is unprovable in $\mathop{\rm ZFC}$. If there exists a non-measurable $A _ {2}$- set (or an $A _ {2}$- set without the Baire property), then there exists an uncountable $CA$- set that does not contain a perfect subset. If every uncountable $CA$- set contains a perfect subset, then the same is true for every uncountable $A _ {2}$- set (see ). The results mentioned here are true not only for the space $I$ but also for the number axis and, in general, for any complete separable metric space. The following theorem on the topological invariance of projective sets is true: The homeomorphic image of a projective set of a given class situated in the same or in any other complete separable metric space is a projective set of the same class.

How to Cite This Entry:
Projective set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_set&oldid=48326
This article was adapted from an original article by A.G. El'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article