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Countable Dense Homogeneity and the Double Arrow Space (1809.06816v1)
Published 18 Sep 2018 in math.GN
Abstract: 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.