Identify the \reftext{def:real-numbers-c54-2026c}{real line} $\mathbb{R}$ with the \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^1$. The \textbf{Borel $\sigma$-algebra} on $\mathbb{R}$, denoted $\mathcal{B}(\mathbb{R})$, is the \reftext{def:generated-sigma-algebra-2026a}{$\sigma$-algebra generated} by the family of all \reftext{def:open-subset-euclidean-space-2026a}{open subsets} of $\mathbb{R}$. Its members are called \textbf{Borel sets}.

In particular every open \reftext{def:interval-real-line-c54-2026c}{interval} is a Borel set; every closed subset is a Borel set (as the complement of an open set); and every interval of any kind is a Borel set, being an intersection of an open set with at most two closed sets.