For any topological space $X$, the Borel sigma algebra of $X$ is the $\sigma$–algebra $\mathcal{B}$ generated by the open sets of $X$. In other words, the Borel sigma algebra is equal to the intersection of all sigma algebras $\mathcal{A}$ of $X$ having the property that every open set of $X$ is an element of $\mathcal{A}$.

An element of $\mathcal{B}$ is called a Borel subset of $X$, or a Borel set.