Partitions of 2ω2ω and completely ultrametrizable spaces

We prove that, for every n  , the topological space ωnω (where ωnωn has the discrete topology) can be partitioned into ℵnℵn copies of the Baire space. Using this fact, we then prove two new theorems about completely ultrametrizable spaces. We say that Y is a condensation of X   if there is a continuous bijection f:X→Yf:X→Y. First, it is proved that ωωωω is a condensation of ωnω if and only if ωωωω can be partitioned into ℵnℵn Borel sets, and some consistency results are given regarding such partitions. It is also proved that it is consistent with ZFC   that, for any n<ωn<ω, c=ωnc=ωn and there are exactly n+3n+3 similarity types of perfect completely ultrametrizable spaces of size cc. These results answer two questions of the first author from [1].