Borelhierarkin

The Borel hierarchy is a classification of subsets of a topological space, particularly the real numbers, based on their complexity. It was introduced by Émile Borel in the early 20th century. The hierarchy is defined recursively, with each level consisting of sets that can be constructed from sets of lower levels using countable unions, intersections, and complements.

The first level of the Borel hierarchy, denoted as Σ1, consists of open sets. The second level, denoted as Σ2, consists of sets that can be expressed as countable unions of closed sets. The third level, denoted as Σ3, consists of sets that can be expressed as countable unions of Σ2 sets, and so on. Similarly, the first level of the dual hierarchy, denoted as Π1, consists of closed sets, the second level, denoted as Π2, consists of sets that can be expressed as countable intersections of open sets, and so on.

The Borel hierarchy is a proper hierarchy, meaning that each level contains sets that are not in the previous level. It is also a complete hierarchy, meaning that every Borel set belongs to some level of the hierarchy. The Borel hierarchy is closely related to the projective hierarchy, which classifies sets based on their definability in the language of second-order arithmetic.

The Borel hierarchy has important applications in various areas of mathematics, including topology, measure theory, and descriptive set theory. It provides a way to classify sets based on their complexity and to study the properties of sets that belong to different levels of the hierarchy.