Borel set

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 $O$.