  Let $\mathbb{A}$ denote the Alexandroff-Urysohn double arrow space. We prove
the following results: (a) $\mathbb{A}\times{}^\omega{2}$ is not countable
dense homogeneous; (b) ${}^{\omega}{\mathbb{A}}$ is not countable dense
homogeneous; (c) $\mathbb{A}$ has exactly $\mathfrak{c}$ types of countable
dense subsets. These results answer questions by Arhangel'ski\u\i, Hru\v{s}\'ak
and van Mill.
