Title: Partitions of 2^omega and Completely Ultrametrizable Spaces
Authors: William R. Brian and Arnold W. Miller
Abstract:
We prove that, for every n , the topological space omega_n^omega
(where omega_n has the discrete topology) can be partitioned into
aleph_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
f:X->Y. First, it is proved that omega^omega is a condensation
of omega_n^omega if and only if omega^omega can be partitioned
into aleph_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 , continuum = 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 cite{Brn}.