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I am looking for some (hopefully not too complicated) examples of zero-dimensional separable metrizable spaces which are:

  • $\sigma$-complete (i.e. can be written as a countable union of completely metrizable subspaces)
  • not strongly $\sigma$-complete (i.e. cannot be written as a countable union of closed completely metrizable subspaces)

EDIT: A Cantor set minus a dense copy of $\mathbb Q ^\omega$ is an example, as noted in a comment below, and this space has certain universal properties. Now I am able to refine my question.

Question. Is the space described above the topologically unique zero-dimensional separable metrizable space which is second category (non-meagre), $\sigma$-complete, and nowhere strongly $\sigma$-complete?

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    $\begingroup$ Take any countable dense set $Q$ in the Cantor set $C$ and consider the space $C^\omega\setminus Q^\omega$. It is $\sigma$-complete but not strongly $\sigma$-complete. In fact, this space contains a closed topological copy of every $\sigma$-complete zero-dimensional space. $\endgroup$ – Taras Banakh yesterday
  • $\begingroup$ @TarasBanakh Perfect! Are you aware of any other information about that space (characterizations, etc)? Is it featured anywhere in the literature? $\endgroup$ – D.S. Lipham yesterday
  • $\begingroup$ Yes, $C^\omega\setminus Q^\omega$ has the following topological characterization: it is a unique meager $\sigma$-complete space which is nowhere absolute $F_{\sigma\delta}$. This characterization belongs to van Engelen (I hope). $\endgroup$ – Taras Banakh yesterday
  • $\begingroup$ @TarasBanakh I think it is not meager though. Did you mean comeager/residual? $\endgroup$ – D.S. Lipham yesterday
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    $\begingroup$ Yes, for the space $C^\omega\setminus Q^\omega$ is should be comeager. But for the space $(C^\omega\times Q^\omega)\times Q$ which is also $\sigma$-complete and nowhere absoute $F_{\sigma\delta}$, the characterization has the meager condition. $\endgroup$ – Taras Banakh 22 hours ago

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