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.