Seven generating families for the Borel sigma-algebra on the real line

Open and closed sets generate the same sigma-algebra by complementation. Open intervals are open, while [L2] expresses every open set using rational open intervals; hence the open intervals and the rational open intervals each generate the sigma-algebra in [L1].

For $a<b$, rational density [L3] gives $(a,b]=\bigcap \{(a,q):q\in \mathbb{Q},q>b\}$ and $(a,b)=\bigcup \{(a,q]:q\in \mathbb{Q},a<q<b\}$. It also gives $(a,\mathrm{\infty })=\bigcup \{(a,q]:q\in \mathbb{Q},q>a\}$ and $(a,\mathrm{\infty })=\bigcup \{(q,\mathrm{\infty }):q\in \mathbb{Q},q>a\}$. Finally $(-\mathrm{\infty },b]=\mathbb{R}\setminus (b,\mathrm{\infty })$ and $(a,b)=(a,\mathrm{\infty })\cap {\bigcup }_{q<b,q\in \mathbb{Q}}(-\mathrm{\infty },q]$. By [L5] all displayed rational-indexed unions and intersections are countable, so these identities give both generator inclusions for the half-open, real-ray, and rational-ray families.

Applying [L4] to the inclusions in steps 1.1 and 2.1 shows that every displayed family generates $\mathcal{B}(\mathbb{R})$.