The Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ is generated by any of the following families:

- (i) the open sets of $\mathbb{R}$, - (ii) the closed sets of $\mathbb{R}$, - (iii) the open intervals $(a, b)$ with $a < b$ in $\mathbb{R}$, - (iv) the half-open intervals $(a, b]$ with $a < b$ in $\mathbb{R}$, - (v) the open rays $(a, \infty)$ with $a \in \mathbb{R}$, - (vi) the closed rays $[a, \infty)$ with $a \in \mathbb{R}$, - (vii) the open rays $(-\infty, a)$ with $a \in \mathbb{R}$.

Moreover, it suffices to take $a, b \in \mathbb{Q}$ in families (iii)-(vii): the $\sigma$-algebra generated by open intervals with rational endpoints equals $\mathcal{B}(\mathbb{R})$.