TITLE: From Counterexamples in Topology, question on the lower limit topology
QUESTION [8 upvotes]: 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 $\mathbb{R}$?

REPLY [8 votes]: A topological space is separable if there exists a countable dense subset. In our case we claim that $\mathbb Q \subset \mathbb R_l$ is a dense subset. Note that the topology on $\mathbb R_l$ is generated by the open sets $[a,b)$ where $a,b \in \mathbb R$ and $a<b$. So it suffices to show that $\mathbb Q \cap [a,b) \neq \emptyset$. This is evident because we know that $\mathbb Q \cap (a,b) \neq \emptyset$ when $a<b$. 
What might have confused you is the fact that the lower limit topology isn't metrizable. So being separable doesn't imply that $\mathbb R_l$ is second countable and in fact it isn't.