Countable dense homogeneity and λ-sets

Rodrigo Hernández-Gutiérrez, Michael Hrušák, and Jan van Mill. "Countable dense homogeneity and λ-sets." Fundamenta Mathematicae 226.2 (2014): 157-172. <http://eudml.org/doc/286141>.

@article{RodrigoHernández2014,
abstract = {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.},
author = {Rodrigo Hernández-Gutiérrez, Michael Hrušák, Jan van Mill},
journal = {Fundamenta Mathematicae},
keywords = {countable dense homogeneous; -set},
language = {eng},
number = {2},
pages = {157-172},
title = {Countable dense homogeneity and λ-sets},
url = {http://eudml.org/doc/286141},
volume = {226},
year = {2014},
}

TY - JOUR
AU - Rodrigo Hernández-Gutiérrez
AU - Michael Hrušák
AU - Jan van Mill
TI - Countable dense homogeneity and λ-sets
JO - Fundamenta Mathematicae
PY - 2014
VL - 226
IS - 2
SP - 157
EP - 172
AB - 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.
LA - eng
KW - countable dense homogeneous; -set
UR - http://eudml.org/doc/286141
ER -