Countable dense homogeneity and $\lambda$-sets

TL;DR

This paper explores the properties of countable dense homogeneity in metric spaces related to $ ext{lambda}$-sets, establishing new connections and examples in topology, including the existence of non-metrizable $ ext{CDH}$ spaces in ZFC.

Contribution

It proves that certain $ ext{lambda}$-sets are $ ext{CDH}$, constructs $ ext{CDH}$ metric spaces of various sizes, and provides the first ZFC example of a compact non-metrizable $ ext{CDH}$ space.

Findings

All sufficiently nice $ ext{lambda}$-sets are $ ext{CDH}$.

Existence of $ ext{CDH}$ metric spaces of size $ ext{cardinality} \kappa ext{ for } ext{uncountable } ext{cardinality} ext{ up to } ext{b}$.

Existence of a non-metrizable compact $ ext{CDH}$ space in ZFC.

Abstract

We show that all sufficiently nice $\lambda$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $\kappa \le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $\kappa$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $\kappa$ is equivalent to the existence of a $\lambda$-set of size $\kappa$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $\omega_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable $\mathsf{CDH}$ space in ZFC.