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An $L$-space is a hereditarily Lindelof regular space which is not separable.

A space is $d$-separable if it contains a dense set which is the countable union of discrete sets.

An $L$-space can't be $d$-separable, because it has no uncountable discrete subsets, however, by a small modification of his celebrated construction of a ZFC $L$-space, Justin Moore provided a ZFC example of an $L$-space with a $d$-separable square.

Moore, Justin Tatch, An $L$ space with a $d$-separable square, Topology Appl. 155, No. 4, 304-307 (2008). ZBL1146.54015.

At first glance $d$-separability looks like the strongest "separability-type" property you could hope to get in the square of an $L$-space, or is it? Consider, for example, the following "selective version" of $d$-separability:

A space is called $D$-separable if, for every sequence $\{D_n: n < \omega \}$ of dense subsets of $X$, there are discrete sets $E_n \subset D_n$, for every $n<\omega$, such that $\bigcup \{E_n: n < \omega \}$ is dense.

The above property lies between a property that the square of an $L$-space clearly can't have (a $\sigma$-disjoint $\pi$-base) and a property that the square of an $L$-space can have ($d$-separability).

QUESTION: Is there an $L$-space with a $D$-separable square?

For more information about $D$-separability see: Bella, Angelo; Matveev, Mikhail; Spadaro, Santi, Variations of selective separability II: Discrete sets and the influence of convergence and maximality, Topology Appl. 159, No. 1, 253-271 (2012). ZBL1239.54014.).

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