Editing Borel set (section)

{{Short description|Class of mathematical sets}}
In [[mathematics]], a '''Borel set''' is any subset of a [[topological space]] that can be formed from its [[open set]]s (or, equivalently, from [[closed set]]s) through the operations of [[countable]] [[union (set theory)|union]], countable [[intersection (set theory)|intersection]], and [[relative complement]].  Borel sets are named after [[Émile Borel]].

For a topological space ''X'', the collection of all Borel sets on ''X'' forms a [[sigma-algebra|&sigma;-algebra]], known as the '''Borel algebra''' or '''Borel &sigma;-algebra'''.  The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).

Borel sets are important in [[measure theory]], since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a [[Borel measure]].  Borel sets and the associated [[Borel hierarchy]] also play a fundamental role in [[descriptive set theory]].

In some contexts, Borel sets are defined to be generated by the [[compact set]]s of the topological space, rather than the open sets. The two definitions are equivalent for many [[well-behaved]] spaces, including all [[Hausdorff space|Hausdorff]] [[σ-compact space]]s, but can be different in more [[pathological (mathematics)|pathological]] spaces.