Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

Abstract

Abstract. We show that, for a coanalytic subspace X of 2 ω , the countable dense homogeneity of X ω 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 ω such that X ω is countable dense homogeneous. This gives the first 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 ω is countable dense homogeneous.

Infinite Powers and Cohen Reals

“…Proof. By [Me3,Propositions 3.4 and 3.3], it follows that either X has a dense completely metrizable subspace or X is not Baire. In the first case, X ω will have a completely metrizable dense subspace as well.…”

“…In the first case, X ω will have a completely metrizable dense subspace as well. In the second case, it is easy to see that X ω will be meager (see for example [Me3,proof of Proposition 4.4]). The proof is concluded by observing that (X ω ) ω is homeomorphic to X ω .…”

Zero-dimensional CDH spaces with a dense completely metrizable subset

“…It is therefore of interest to construct a non-complete CDHspace with a dense complete subset. It was realized by R. Hernández-Gutiérrez, M. Hrušák, and J. van Mill [2, Corollary 4.6] and A. Medini [3, Theorem 7.3]. They shown that if a homogeneous space X with a dense complete subset can be embedded in the Cantor set 2 ω by a special way, then X is CDH.…”

“…A separable topological space X is countable dense homogeneous (briefly, CDH) if, given any two countable dense subsets A and B of X, there is a homeomorphism h : X → X such that h(A) = B. Similarly, a space X is densely homogeneous (briefly, DH) if, given any two σ-discrete dense subsets A and B of X, there is a homeomorphism h : X → X such that h(A) = B.Medini [3] says that a separable space X is countably controlled if for every countable D ⊂ X there exists a Polish subspace G of X such that D ⊂ G. Similarly, we shall say that a space X is σ-discretely controlled if for every σ-discrete D ⊂ X there exists a completely metrizable subspace G of X such that D ⊂ G. Clearly, every σ-discretely controlled space 2010 Mathematics Subject Classification. 54H05, 54E52.…”

“…Medini [3] says that a separable space X is countably controlled if for every countable D ⊂ X there exists a Polish subspace G of X such that D ⊂ G. Similarly, we shall say that a space X is σ-discretely controlled if for every σ-discrete D ⊂ X there exists a completely metrizable subspace G of X such that D ⊂ G. Clearly, every σ-discretely controlled space 2010 Mathematics Subject Classification. 54H05, 54E52.…”