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 Determina...

We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply deened compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the...

We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in discriptive set theoretic (topological) sense and in measure theoretic sense. In particular, it gives a new perspective on Vershik’s theorems on genericity and randomness of Urysohn’s space among separable metric spaces.

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than k can be always mapped onto a k-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application we essentially answer a question of Urbański by showing that the transfinite Hausdorff...