A separable space $X$ is said to be countably dense homogeneous (CDH) if for any two countable dense subsets $A$ and $B$ of $X$ there exists a homeomorphism $f:X\longrightarrow X$ such that $f(A)=B.$
It is well known that the Cantor set is countably dense homogeneous, as it is a strongly locally homogeneous Polish space. My questions are the following:
Does there exist any direct proof of that the Cantor set is CDH?
Can anyone suggest a textbook discussing CDH spaces in detail?