Partitions of 2^{\omega} and completely ultrametrizable spaces

TL;DR

This paper explores partitions of certain topological spaces into copies of the Baire space and investigates properties of completely ultrametrizable spaces, including their condensations and the diversity of their similarity types under various set-theoretic assumptions.

Contribution

It establishes new partition results for spaces like {omega}_n^{omega} and characterizes when the Baire space is a condensation of these spaces, also analyzing the number of similarity types of ultrametrizable spaces.

Findings

Partition of {omega}_n^{omega} into {omega}_n copies of Baire space

Characterization of when the Baire space is a condensation of {omega}_n^{omega}

Consistency results on the number of similarity types of ultrametrizable spaces

Abstract

We prove that, for every n, the topological space {\omega}_n^{\omega} (where {\omega}_n has the discrete topology) can be partitioned into {\omega}_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 {\omega}_n^{\omega} if and only if it can be partitioned into {\omega}_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 < {\omega}, the continuum is {\omega}_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.