Countable dense homogeneity and the double arrow space

Let A denote the Alexandroff–Urysohn double arrow space. We prove the following results: (a) A × 2 ω is not countable dense homogeneous; (b) A ω is not countable dense homogeneous; (c) A has exactly c types of countable dense subsets. These results answer questions by Arhangelʼskiĭ, Hrušák and van Mill.