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Let us say a topological space $X$ is a countable union of second countable spaces if there exists a sequence of subsets $\{X_n\}$ of $X$ with $X=\cup X_n$ such that the relative topology on $X_n$'s is second countable. Clearly $X$ will be separable.

Q. What topological properties P are strictly between separabilty and countable union of second countable spaces?

Countable union of $2^{ed}$-countable spaces $\prec$ P $\prec$ Separablity

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    $\begingroup$ Hereditarily separable? Hereditarily separable and hereditarily Lindelof? $\endgroup$ – bof May 21 '18 at 7:45
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    $\begingroup$ Having countable network weight (or equivalently, being the continuous image of a separable metric space). $\endgroup$ – Ramiro de la Vega May 21 '18 at 18:30
  • $\begingroup$ @RamirodelaVega this is called a “cosmic” space, IIRC. $\endgroup$ – Henno Brandsma May 22 '18 at 4:40

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