A space $X$ is *discretely Lindelöf* iff given any discrete subset $D$ of $X$, its closure in $X$ is Lindelöf. Such spaces were introduced by Arkhangel'skii about 15 years ago (if I am not mistaken) under the name "strongly discretely Lindelöf", but after browsing more recent articles it seems that "strongly" has been dropped. It is an open problem whether there is a regular or Tychonov discretely Lindelöf non-Lindelöf space (and apparently the Hausdorff case is also open).

My question is: Is it known whether the product of a discretely Lindelöf space with [0,1] (or with $\omega +1$) is discretely Lindelöf ?

To have a positive answer, I think it would be enough to prove that the closure in $X$ of a countable union of discrete subspaces of $X$ is Lindelöf, but I could not go beyond this.