Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

Medini, Andrea

Canadian mathematical bulletin, Tome 58 (2015) no. 2, pp. 334-349

Résumé

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.

Détail

DOI : 10.4153/CMB-2014-062-6

Mots-clés : 54H05, 54G20, 54E52, countable dense homogeneous, infinite power, coanalytic, Polish, λ'-set