Countable Dense Homogeneity of Definable Spaces

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 the following are equivalent: (1) X is Gδ in 2 ω , (2) X is CDH and (3) X is homeomorphic to 2 or to ω . 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 large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Steprāns and Zhou by showing that p = min{κ : 2 is not CDH}.