Borel hierarchy

In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory.

One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank. Properties of sets of small finite ranks are important in measure theory and analysis.

The Borel algebra in an arbitrary topological space is the smallest collection of subsets of the space that contains the open sets and is closed under countable unions and complementation. It can be shown that the Borel algebra is closed under countable intersections as well.

A short proof that the Borel algebra is well-defined proceeds by showing that the entire powerset of the space is closed under complements and countable unions, and thus the Borel algebra is the intersection of all families of subsets of the space that have these closure properties. This proof does not give a simple procedure for determining whether a set is Borel. A motivation for the Borel hierarchy is to provide a more explicit characterization of the Borel sets.

The Borel hierarchy or boldface Borel hierarchy on a space X consists of classes ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$, ${\displaystyle \mathbf {\Pi } _{\alpha }^{0}}$, and ${\displaystyle \mathbf {\Delta } _{\alpha }^{0}}$ for every countable ordinal ${\displaystyle \alpha }$ greater than zero. Each of these classes consists of subsets of X. The classes are defined inductively from the following rules:

The motivation for the hierarchy is to follow the way in which a Borel set could be constructed from open sets using complementation and countable unions. A Borel set is said to have finite rank if it is in ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$ for some finite ordinal ${\displaystyle \alpha }$; otherwise it has infinite rank.

If ${\displaystyle \mathbf {\Sigma } _{1}^{0}\subseteq \mathbf {\Sigma } _{2}^{0}}$ then the hierarchy can be shown to have the following properties:

The classes of small rank are known by alternate names in classical descriptive set theory.