Countable dense homogeneity and topological groups

TL;DR

This paper constructs a $ZFC$ example of a non-Polish topological group that is countable dense homogeneous ($\\mathsf{CDH}$) by building a dense subgroup $X\\subseteq \\mathbb{Z}^\\omega$ of size $\\mathfrak{b}$ that is a $\\lambda$-set. The construction uses a transfinite recursion to produce $f_\\alpha$ with positivity and domination properties, defines $Q=\\{z\\in \\mathbb{Z}^\\omega: z=^* \\vec{0}\\}$ and $X=\\langle \\{f_\\alpha: \\alpha<\\mathfrak{b}\\}\\cup Q\\rangle$, and then shows $X$ is a $\\lambda$-set via a Rothberger-type argument; Medvedev-type results then yield that $X$ is $\\mathsf{CDH}$ and not locally precompact, giving a non-Polish example in $\\mathsf{ZFC}$. The main contribution is the first $ZFC$ construction of an uncountable $\\lambda$-set that yields a non-Polish $\\mathsf{CDH}$ group, complemented by a conjecture about Baire $\\mathsf{CDH}$ groups and a proved special-case when the group has index $2$ in its completion. These results connect definability properties (being a $\\lambda$-set) with topological group structure and guide future work on the Baire behavior of CDH groups.

Abstract

Building on results of Medvedev, we construct a $\mathsf{ZFC}$ example of a non-Polish topological group that is countable dense homogeneous. Our example is a dense subgroup of $\mathbb{Z}^ω$ of size $\mathfrak{b}$ that is a $λ$-set. We also conjecture that every countable dense homogenous Baire topological group with no isolated points contains a copy of the Cantor set, and give a proof in a very special case.

Theorems & Definitions (12)

• Lemma 2.1: Rothberger
• Theorem 3.1: Medvedev
• Theorem 3.2: Medvedev
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Conjecture 4.2
• Lemma 4.3
• proof