In mathematics, a Borel set is a set that belongs to the σ-algebra generated by the open sets of a topological space. Thus, every open set is a Borel set, as are countable unions of open sets (i.e., unions of countably many open sets), and countable intersections of countable unions of open sets, etc.
Let be a topological space, i.e. is a set and are the open sets of (or, equivalently, the topology of ). Then is a Borel set of if , where denotes the σ-algebra generated by .
The σ-algebra generated by is simply the smallest σ-algebra containing the sets in or, equivalently, the intersection of all σ-algebras containing .