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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.

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    $\begingroup$ Hi Mathieu, you might be interested in this paper by Peng and Tall [topology.auburn.edu/tp/reprints/v32/tp32018.pdf] where they make the following two observations: 1) let X be discretely Lindelof space where closures of Lindelof subspaces are Lindelof. Then closures of sigma-discrete subspaces are Lindelof. 2) It follows from the Eisworth-Nyikos trichotomy (a consequence of PFA) that every locally compact normal space where closures of sigma-discrete subspaces are Lindelof is actually Lindelof. $\endgroup$ – Santi Spadaro May 20 '15 at 17:36
  • $\begingroup$ Hi Santi, thanks for your comment. I actually looked at this paper some time ago but had completely forgotten this part. I'll have a closer look when I have some time (in a couple of weeks, unfortunately). $\endgroup$ – Mathieu Baillif May 20 '15 at 20:33

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