The classical Alexandroff-Urysohn double arrow space \(\mathbb{A}\) was first defined in [P. Alexandroff and P. Urysohn, Mémoire sur les espaces topologiques compacts dédié à Monsieur D. Egoroff. Verhandelingen Amsterdam 14, No.1, 93 S. (1929; JFM 55.0960.02)]. Let \(\mathbb{A}_{0}=(0,1]\times\{0 \}\), \(\mathbb{A}_{1}=[0,1)\times\{1 \}\) and \(\mathbb{A}=\mathbb{A}_{0}\cup \mathbb{A}_{1}\). The lexicographic (strict) order on \(\mathbb{A}\) is defined as \(\langle x,t\rangle < \langle y,s\rangle\) if \(x<y\) or \(x=y\) and \(t<s\). Then \(\mathbb{A}\) is given the order topology and it is known that \(\mathbb{A}\) is separable, first countable, compact, 0-dimensional and of weight \(\mathrm{c}\). Moreover, both \(\mathbb{A}_{0}\) and \(\mathbb{A}_{1}\) have the Sorgenfrey line topology as subspaces of \(\mathbb{A}\) and both are dense in \(\mathbb{A}\).
In this paper, the author undertakes to answer the questions in [A. V. Arhangel’skii and J. van Mill, Proc. Am. Math. Soc. 141, No. 11, 4031–4038 (2013; Zbl 1315.54004)] and proves the following results:
i) \(\mathbb{A}\)\(\times\)\(^{\omega}2\) is not countable dense homogeneous (CDH for short), ii) \(^{\omega}\)\(\mathbb{A}\) is not CDH, iii) \(\mathbb{A}\) has exactly \(\mathrm{c}\) types of countable dense subsets.
Moreover, the author shows that \(\mathbb{A}\) does not contain any subspace homeomorphic to \(\mathbb{A}\)\(\times\)\(({\omega}+1)\) and proves if \(C\) is compact and countably infinite then the \(C\)\(\times\)\(\mathbb{A}\) is neither homogeneous nor CDH.