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The Borel Hierarchy is a system for [objectively] deciding which subsets of a Polish space are superior to other ones and must be obeyed. It is defined using transfinite induction, with the base case Σ_1 = { the open sets }, Π_1 = { the closed sets }, and for a countable ordinal α, Σ_α is the collection of sets which are the countable union of sets appearing in Π_β for β < α, and dually, Π_α is the collection of sets which are the countable [intersection] of sets appearing in Σ_β for β < α. The rank of a set is the index of [the first time] the set appears in the hierarchy. The lowest rank sets are the open and closed sets, and the highest rank set is the Borel algebra.
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