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Countable dense homogeneity and $\lambda $-sets

Volume 226 / 2014

Rodrigo Hernández-Gutiérrez, Michael Hrušák, Jan van Mill Fundamenta Mathematicae 226 (2014), 157-172 MSC: Primary 54H05; Secondary 03E15, 54E50. DOI: 10.4064/fm226-2-5

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.

Authors

  • Rodrigo Hernández-GutiérrezDepartment of Mathematics and Statistics
    York University
    Toronto, ON M3J 1P3, Canada
    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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