We show that all sufficiently nice λ-sets are countable dense homogeneous (𝖢𝖣𝖧). From this fact we conclude that for every uncountable cardinal κ ≤ 𝔟 there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every 𝖢𝖣𝖧 metric space has size either ω₁ or 𝔠. An example of a Baire 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 𝖢𝖣𝖧 space in ZFC.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-2-5,
author = {Rodrigo Hern\'andez-Guti\'errez and Michael Hru\v s\'ak and Jan van Mill},
title = {Countable dense homogeneity and $\lambda$-sets},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {157-172},
zbl = {1301.54050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-2-5}
}
Rodrigo Hernández-Gutiérrez; Michael Hrušák; Jan van Mill. Countable dense homogeneity and λ-sets. Fundamenta Mathematicae, Tome 227 (2014) pp. 157-172. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-2-5/