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1. CMB 2015 (vol 58 pp. 334)

Medini, Andrea
 Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces 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 HruÅ¡Ã¡k and Zamora AvilÃ©s. Then, inspired by results of HernÃ¡ndez-GutiÃ©rrez, HruÅ¡Ã¡k 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 HruÅ¡Ã¡k and Zamora AvilÃ©s. 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, $\lambda'$-setCategories:54H05, 54G20, 54E52

2. CMB 2012 (vol 56 pp. 860)

van Mill, Jan
 On Countable Dense and $n$-homogeneity We prove that a connected, countable dense homogeneous space is $n$-homogeneous for every $n$, and strongly 2-homogeneous provided it is locally connected. We also present an example of a connected and countable dense homogeneous space which is not strongly 2-homogeneous. This answers Problem 136 of Watson in the Open Problems in Topology Book in the negative. Keywords:countable dense homogeneous, connected, $n$-homogeneous, strongly $n$-homogeneous, counterexampleCategories:54H15, 54C10, 54F05
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