Countable Dense Homogeneity and the Double Arrow Space

TL;DR

This paper investigates the countable dense homogeneity of the double arrow space and its products, revealing non-homogeneity in certain cases and classifying types of countable dense subsets, thus answering longstanding open questions.

Contribution

It establishes new results on the non-countable dense homogeneity of specific products of the double arrow space and classifies the types of countable dense subsets, addressing open problems.

Findings

$ ext{A} imes {}^ ext{ω} 2$ is not countable dense homogeneous.

${}^ ext{ω} ext{A}$ is not countable dense homogeneous.

$ ext{A}$ has exactly $ ext{c}$ types of countable dense subsets.

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.