Borel Sigma-Algebra on the Real Line

Identify the real line $\mathbb{R}$ with the Euclidean space $\mathbb{R}^1$. The Borel $\sigma$-algebra on $\mathbb{R}$, denoted $\mathcal{B}(\mathbb{R})$, is the $\sigma$-algebra generated by the family of all open subsets of $\mathbb{R}$. Its members are called Borel sets.

In particular every open 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.