On page 75 of Counterexamples in Topology, the author writes that the lower limit topology on $\mathbb{R}$ is separable since $\mathbb{Q}$ is dense in $\mathbb{R}$.
Could someone offer more detail on why this is so? I know $\overline{\mathbb{Q}}=\mathbb{R}$ in the standard topology, but might there be something different going on since the closure of $\mathbb{Q}$ might be proven differently depending on the topology of $\mathbb{R}$?