Countable dense homogeneity and $\lambda $-sets

Rodrigo Hernández-Gutiérrez ¹   ; Michael Hrušák ²   ; Jan van Mill ³

¹ Department of Mathematics and Statistics York University Toronto, ON M3J 1P3, Canada
² Centro de Ciencias Matemáticas UNAM A.P. 61-3 Xangari Morelia, Michoacán 58089, México
³ Faculty 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

Fundamenta Mathematicae, Tome 226 (2014) no. 2, pp. 157-172

Résumé

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.

Détail

DOI : 10.4064/fm226-2-5

Keywords: sufficiently nice lambda sets countable dense homogeneous mathsf cdh conclude every uncountable cardinal kappa mathfrak there countable dense homogeneous metric space size kappa moreover existence meager itself countable dense homogeneous metric space size kappa equivalent existence lambda set size kappa other consistent continuum arbitrarily large every mathsf cdh metric space has size either omega nbsp mathfrak example baire mathsf cdh metric space which completely metrizable presented finally answering question arhangelskii van mill that there compact non metrizable mathsf cdh space zfc