Countable dense homogeneity and λ-sets

• Autorzy: Rodrigo Hernández-Gutiérrez , Michael Hrušák , Jan van Mill
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 03E15: Descriptive set theory
• 54E50: Complete metric spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 226
• Numer: 2
• Strony: 157-172

Daty

wydano

2014

Twórcy

autor

Rodrigo Hernández-Gutiérrez

• Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada

autor

Michael Hrušák

• Centro de Ciencias Matemáticas, UNAM, A.P. 61-3 Xangari, Morelia, Michoacán 58089, México

autor

Jan van Mill

• Faculty of Sciences, VU University Amsterdam, De Boelelaan 1081A, 1081 HV Amsterdam, The Netherlands
• Faculty of Electrical Engineering, Mathematics and Computer Science, TU Delft, Postbus 5031, 2600 GA Delft, The Netherlands
• Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South Africa

Identyfikatory

DOI

10.4064/fm226-2-5