A topological space $X$ is weakly Lindelof if every open cover has a countable subfamily $U$ such that $\bigcup \{ V: V\in U\}$ is dense in X.


Question: Are closed subsets of weakly Lindelof spaces necessarily weakly Lindelof?



closed subset of weakly lindelof
The [Niemytzki plane](https://en.wikipedia.org/wiki/Moore_plane) is weakly Lindelöf (the open upper half plane is a dense Lindelöf subspace); the $x$-axis is an uncountable closed and discrete subspace.


