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

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 conden...

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Bibliographic Details
Published in:arXiv.org 2014-06
Main Authors: Brian, William R, Miller, Arnold W
Format: Article
Language:English
Subjects:
Borel sets
Condensation
Partitions
Topology
Online Access:Get full text
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Summary: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.
ISSN:2331-8422