Borel $\sigma$-algebra

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

An element of $ℬ$ is called a Borel subset of $X$, or a Borel set.