I am completing my undergrad thesis about topological properties of some subspaces of the real numbers, and CDH spaces are one of the topics I´ve covered (I know almost nothing about it, I only prove that the Cantor space, the set of irrational numbers and $\mathbb{R}$ are CDH). However, I think I was able to prove a stronger version of countable dense homogeneity for the Cantor set. Specifically:
Let $A_n$ be a sequence of disjoint countable dense subspaces of the Cantor set $2^\omega$, and let $B_n$ be other such sequence. Then there is an homeomorphism $f:2^\omega\to2^\omega$ such that $f(A_n)=B_n$ for all $n$.
If this property is true for $2^\omega$, it is not difficult to prove that it is also true for the Baire space $\omega^\omega$, and I think I was also able to find other proof that $\mathbb{R}$ has this property too (probably manifolds will have it too but that´s outside the scope of my thesis). I was going to include this property and it would be useful to reference some paper about the topic, but I found nothing about it in a few papers I looked up about CDH spaces.
So my question is, is there any paper/book where I can find information about this property, or about any other similar properties or concepts that are easily seen to imply this one?
