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?

  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.