Borel set

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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.

Formal definition

Let (X,O) be a topological space, i.e. X is a set and O are the open sets of X (or, equivalently, the topology of X). Then A \subset X is a Borel set of X if A \in \sigma(O) , where \sigma(O) denotes the σ-algebra generated by O.

The σ-algebra generated by O is simply the smallest σ-algebra containing the sets in O or, equivalently, the intersection of all σ-algebras containing O.

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