Countable dense homogeneity and $\lambda $-sets
Volume 226 / 2014
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 $\mathfrak c$. 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.
