Countable dense homogeneity and the double arrow space

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