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Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

Published online by Cambridge University Press:  20 November 2018

Andrea Medini
Andrea Medini*
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, WGähringer Straße 25, A-1090 Wien, Austria. e-mail: andrea.medini@univie.ac.at
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Abstract

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We show that for a coanalytic subspace $X$ of ${{2}^{\omega }}$, the countable dense homogeneity of ${{X}^{\omega }}$ is equivalent to $X$ being Polish. This strengthens a result of Hrušák and Zamora Avilés. Then, inspired by results of Hernández-Gutiérrez, Hrušák, and van Mill, using a technique of Medvedev, we construct a non-Polish subspace $X$ of ${{2}^{\omega }}$ such that ${{X}^{\omega }}$ is countable dense homogeneous. This gives the first $\text{ZFC}$ answer to a question of Hrušák and Zamora Avilés. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a question of Medini and Milovich. Finally, we show that if every countable subset of a zero-dimensional separable metrizable space $X$ is included in a Polish subspace of $X$, then ${{X}^{\omega }}$ is countable dense homogeneous.

Keywords

54H0554G2054E52countable dense homogeneousinfinite powercoanalyticPolishλ'-set
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 334 - 349
Copyright
Copyright © Canadian Mathematical Society 2015

References

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