I'm working on the study of almost Lindelöf spaces and I'm stuck searching a counterexample. First, the definition.
Let $X$ be a topological space. We say that $X$ is an almost Lindelöf space if for every open cover $\mathcal{U}$ of $X$ there exist a countable subset $\mathcal{U}_0$ of $\mathcal{U}$ such that $$X=\bigcup_{U\in\mathcal{U}_0} \text{cl}(U)$$
In general, the property doesn't preserve to closed subspaces. The example is $\kappa\mathbb{N}$ (the Katetov extension of $\mathbb{N}$) because is $H-$closed (and therefore almost Lindelöf) and $\kappa\mathbb{N}\setminus\mathbb{N}$ is a uncountable closed discrete subspace. My question is: is there another topological space different from Katetov extension that proves that the property of being almost Lindelöf doesn't preserve by take closed subspaces? I really appreciate any help you can provide me.
Example 3.3 in this overview paper is almost Lindelöf with an uncountable discrete closed subspace.