1
$\begingroup$

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

$\endgroup$
3
  • 1
    $\begingroup$ Hereditarily separable? Hereditarily separable and hereditarily Lindelof? $\endgroup$
    – bof
    May 21, 2018 at 7:45
  • 1
    $\begingroup$ Having countable network weight (or equivalently, being the continuous image of a separable metric space). $\endgroup$ May 21, 2018 at 18:30
  • $\begingroup$ @RamirodelaVega this is called a “cosmic” space, IIRC. $\endgroup$ May 22, 2018 at 4:40

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.