Consider the space $\Bbb R_l$ that is $\Bbb R$ with the lower limit topology. Show that $\Bbb R_l$ is Lindelöf.
I am trying to understand a proof, but I don't know why the result implies that $\Bbb R_l$ is Lindelöf. It goes like this.
Let $\mathcal{A}=\{U_j \mid j \in J\}$ be an open cover of $\Bbb R_l$. Let $V_j=\operatorname{int}(U_j)$ in the standard topology of $\Bbb R$ and $V=\bigcup\{V_j \mid j \in J\}$. Since $V$ is second-countable and therefore Lindelöf in the standard topology of $\Bbb R$ we know that there exists countable $K\subset J$ such that $V=\bigcup\{V_j \mid j \in K\}$ covers $\Bbb R$.
Showing that $A= \Bbb R \setminus V$ is countable will imply the result.
The proof ends here and I cannot figure out why showing $A= \Bbb R \setminus V$ will imply that $\Bbb R_l$ is Lindelöf?
