Abstract view
Countable Dense Homogeneity in Powers of Zerodimensional Definable Spaces


Published:20150211
Printed: Jun 2015
Andrea Medini,
Kurt Gödel Research Center for Mathematical Logic , University of Vienna , Währinger Straße 25 , A1090 Wien, Austria
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 HernándezGutiérrez,
Hrušák and van Mill, using a technique of Medvedev, we
construct a nonPolish 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 zerodimensional separable metrizable space $X$ is
included in a Polish subspace of $X$ then $X^\omega$ is countable
dense homogeneous.
MSC Classifications: 
54H05, 54G20, 54E52 show english descriptions
Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] Counterexamples Baire category, Baire spaces
54H05  Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] 54G20  Counterexamples 54E52  Baire category, Baire spaces
