Let $X$ be a compact space and $L$ is the smallest family of subspaces of$\,X\,$that contains all closed sets and is closed with respect to countable union and intersection. The question is :-
Is all element of $L$ have Lindelöf property?
Remark:- Lindelöf property and Lindelöf space are same, since the definition of Lindelöf depend of regularity, I want to just add that every Regular space is Hausdorff.
Update:- We know that Lindelöf Property is hereditary with respect to $F_\sigma{-set}$. So I think this fact + the space been compact will be the key to the solution.