The lower-limit line has a clopen basis, is regular, and is Lindelöf under countable choice

Each $[a,b)$ is clopen: its complement is $(-\mathrm{\infty },a)\cup [b,\mathrm{\infty })$, a union of lower-limit basic intervals. Hence the line is regular by [L1].

Let $\mathcal{U}$ be an open cover and let $O$ be the union of the usual intervals $(a,b)$ for which $[a,b)$ lies in a member of $\mathcal{U}$. The rational-endpoint intervals $[p,q)$ contained in members of $\mathcal{U}$ cover $O$; they form an at most countable family by [L2].

Put $D=\mathbb{R}\setminus O$. For $x\in D$, some $[x,b_x)$ lies in a member of $\mathcal{U}$, and $[x,b_x)\cap D=\{x\}$. The first rational $q_x\in (x,b_x)$ in a fixed enumeration exists, and the intervals $(x,q_x)$ are pairwise disjoint; their first rationals $r_x$ are therefore distinct. Thus $x\mapsto r_x$ injects $D$ into $\mathbb{Q}$, so $D$ is at most countable.

By [A1], choose one member of $\mathcal{U}$ covering each point of the at most countable set $D$. Together with one covering member for each rational-endpoint interval used in step 1.2, these form an at most countable subcover of $\mathcal{U}$.

Therefore the lower-limit line is Lindelöf under countable choice.