Countable Dense Homogeneity and the Double Arrow Space

Abstract

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