Countable Basis of Real Number Line

Theorem
Let $\struct {\R, \tau_d}$ be the real number line considered as a topological space under the usual (Euclidean) topology.

Let $\BB$ be the set of subsets of $\R$ defined as:
 * $\BB = \set {\openint a b: a, b \in \Q,\ a < b}$

That is, $\BB$ is the set of open intervals of $\R$ whose endpoints are rational numbers.

Then $\BB$ forms a countable basis of $\struct {\R, \tau_d}$