Partitions of 2^ω and completely ultrametrizable spaces

Abstract

We prove that, for every n, the topological space ω_n^ω (where ω_n has the discrete topology) can be partitioned into ω_n copies of the Baire space. Using this fact, the authors then prove two new theorems about completely ultrametrizable spaces. We say that Y is a condensation of X if there is a continuous bijection from X to Y. First, it is proved that the Baire space is a condensation of ω_n^ω if and only if it can be partitioned into ω_n 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 < ω, the continuum is ω_n and there are exactly n+3 similarity types of perfect completely ultrametrizable spaces of size continuum. These results answer two questions of the first author from a previous paper.