This question is part of my Topology assignment and I am stuck on it. So, I am asking for guidance here.
Prove that a regular $(X,\mathcal{T})$ is Lindelöf iff each open cover $\mathcal{O}$ has a countable subset $\mathcal{O}'=\{U_n \mid n \in \mathbb{N}\}$ such that $\{\overline{U_n} \mid U_n \in \mathcal{O}'\}$ covers $X$.
I have proved it lindelöf assuming the given condition but I am unable to think how should I proceed converse because the set of $\overline{U_n}$ covering $X$ are not are not open and I am unable to think how should I find such subcover.
Please guide.