# Suslin theorem

(in descriptive set theory)

There exists an $\mathcal{A}$-set (of the number axis $\mathbb{R}$) that is not a Borel set.

In order that a given $\mathcal{A}$-set be a Borel set it is necessary and sufficient that its complement also be an $\mathcal{A}$-set.

Every $\mathcal{A}$-set in the $n$-dimensional space $\mathbb{R}^n$ is the (orthogonal) projection of a Borel set (even of type $G_\delta$) in $\mathbb{R}^{n+1}$ (and consequently, a plane Borel set of type $G_\delta$ exists whose projection is not a Borel set); the projection of an $\mathcal{A}$-set is an $\mathcal{A}$-set.

All these results were obtained by M.Ya. Suslin . In order to define an $\mathcal{A}$-set, he used the $\mathcal{A}$-operation, while other methods of defining $\mathcal{A}$-sets were discovered subsequently. The $\mathcal{A}$-operation was in fact first discovered by P.S. Aleksandrov , who demonstrated (although he did not explicitly formulate it) that every Borel set can be obtained as the result of the $\mathcal{A}$-operation over closed sets (and is consequently an $\mathcal{A}$-set), and used this to prove a theorem on the cardinality of Borel sets (in fact, of $\mathcal{A}$-sets) in $\mathbb{R}$. N.N. Luzin subsequently posed the question of the existence of an $\mathcal{A}$-set that is not a Borel set. Theorem 1) answered this question. Theorems 1) and 2) were put forward by Suslin without proof . Suslin did subsequently prove them, but it was not until Luzin simplified the proofs that they were published. In order to prove 1), Suslin constructed a plane $\mathcal{A}$-set that was universal for all Borel sets and examined the set of its points that lie on the diagonal $x=y$ (see , p. 94). Theorem 2) is now often called the Suslin criterion for Borel sets. Suslin's proof of this theorem was based on a decomposition of a $\mathcal{CA}$-set into the sum of $\aleph_1$ Borel sets (see , ).

How to Cite This Entry:
Suslin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Suslin_theorem&oldid=33668
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