# \$sigma\$-complete zero-dimensional spaces

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)