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# Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

Published:2015-02-11
Printed: Jun 2015
• Andrea Medini,
Kurt Gödel Research Center for Mathematical Logic , University of Vienna , Währinger Straße 25 , A-1090 Wien, Austria
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## Abstract

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 $\mathsf{ZFC}$ 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: countable dense homogeneous, infinite power, coanalytic, Polish, $\lambda'$-set
 MSC Classifications: 54H05 - Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] 54G20 - Counterexamples 54E52 - Baire category, Baire spaces

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