nonBorel

nonBorel is a term used in descriptive set theory to denote subsets of a Polish space that are not Borel sets. Borel sets are the sigma-algebra generated from open sets through countable unions, intersections, and complements; they form the smallest family of sets containing the open sets and closed under these operations.

There exist non-Borel sets because the Borel sigma-algebra has cardinality continuum, while the full power set of the space has a strictly larger cardinality. Consequently, not every subset is Borel, and non-Borel sets occur naturally in the study of definability and measurability on spaces like the real line.

In descriptive set theory, a distinction is drawn between Borel, analytic (Σ1^1), and coanalytic (Π1^1) sets. There are sets that are analytic or coanalytic but not Borel, illustrating that the Borel hierarchy does not exhaust all definable subsets. The Borel hierarchy itself extends through countable ordinals, and the projective hierarchy encompasses broader classes of sets beyond Borel, analytic, and coanalytic.

Constructions of non-Borel sets often involve projections of Borel sets or coding methods that produce subsets not obtainable by the standard Borel operations. Non-Borel sets play a central role in arguments and counterexamples within descriptive set theory, measure theory, and recursion theory.

In practical contexts, many mathematical analyses and probability frameworks focus on Borel or Lebesgue measurable sets. Non-Borel sets are typically studied for foundational and theoretical purposes, illustrating limitations of definability and the boundaries of sigma-algebras.