# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Countable dense homogeneity and $\lambda$-sets

### Tom 226 / 2014

Fundamenta Mathematicae 226 (2014), 157-172 MSC: Primary 54H05; Secondary 03E15, 54E50. DOI: 10.4064/fm226-2-5

#### Streszczenie

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.

#### Autorzy

• Rodrigo Hernández-GutiérrezDepartment of Mathematics and Statistics
York University
e-mail
• Michael HrušákCentro de Ciencias Matemáticas
UNAM
A.P. 61-3 Xangari
Morelia, Michoacán 58089, México
e-mail
• Jan van MillFaculty of Sciences
VU University Amsterdam
De Boelelaan 1081A
1081 HV Amsterdam, The Netherlands
and
Faculty of Electrical Engineering
Mathematics and Computer Science
TU Delft
Postbus 5031
2600 GA Delft, The Netherlands
and
Department of Mathematical Sciences
University of South Africa
P.O. Box 392
0003 Unisa, South Africa
e-mail

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