Borel Hierarchy

The Borel hierarchy or boldface Borel hierarchy on a space X consists of classes, and for every countable ordinal greater than zero. Each of these classes consists of subsets of X. The classes are defined inductively from the following rules:

• A set is if and only if it is open.
• A set is if and only if its complement is .
• A set is for if and only if there is a sequence of sets such that each is for some and .
• A set is if and only if it is both and .

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 for some finite ordinal α; otherwise it has infinite rank.

The hierarchy can be shown to have the following properties:

• . Moreover, a set is in this union if and only if it is Borel.
• For every α, . Thus, once a set is in or, that set will be in all classes in the hierarchy corresponding to ordinals greater than α
• If is an uncountable Polish space, it can be shown that is not contained in for any, and thus the hierarchy does not collapse.