Countable dense homogeneity of definable spaces

Abstract

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω X\subseteq 2^{\omega } the following are equivalent: (1) X X is G δ G_{\delta } in 2 ω 2^{\omega } , (2) X ω X^{\omega } is CDH and (3) X ω X^{\omega } is homeomorphic to 2 ω 2^{\omega } or to ω ω \omega ^{\omega } . Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr a ¯ \bar {\text {a}} ns and Zhou, by showing that p = min { κ : 2 κ \mathfrak {p}= \min \{\kappa : 2^{\kappa } is not CDH } \} .