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Search: All articles in the CMB digital archive with keyword infinite power

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1. CMB 2017 (vol 61 pp. 812)

Medini, Andrea; van Mill, Jan; Zdomskyy, Lyubomyr S.
 Infinite Powers and Cohen Reals We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of $Z^\omega$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of $\omega_1$ Cohen reals, viewed as a subspace of $2^\omega$. Keywords:infinite power, zero-dimensional, first-countable, homogeneous, Cohen real, h-homogeneous, rigidCategories:03E35, 54B10, 54G20

2. 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
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