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?