Israel Journal of Mathematics

, Volume 65, Issue 2, pp 153–164 | Cite as

Countable dense homogeneous spaces under Martin’s axiom

  • Stewart Baldwin
  • Robert E. Beaudoin


We show that Martin’s axiom for countable partial orders implies the existence of a countable dense homogeneous Bernstein subset of the reals. Using Martin’s axiom we derive a characterization of the countable dense homogeneous spaces among the separable metric spaces of cardinality less thanc. Also, we show that Martin’s axiom implies the existence of a subset of the Cantor set which isλ-dense homogeneous for everyλ <c.


Partial Order Dense Subset Clopen Subset Generic Filter Countable Dense Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ba]
    J. Baumgartner,Order types of real numbers and other uncountable orderings, inOrdered Sets (I. Rival, ed.), D. Reidel, Dordrecht, 1982, pp. 239–277.Google Scholar
  2. [Be]
    R. Bennett,Countable dense homogeneous spaces, Fund. Math.74 (1972), 189–194.MathSciNetGoogle Scholar
  3. [FMS]
    M. Foreman, M. Magidor and S. Shelah,Martin’s maximum, saturated ideals, and non-regular ultrafilters, Part 1, to appear.Google Scholar
  4. [FZ]
    B. Fitzpatrick and Zhou Hao-xuan,A note on countable dense homogeneity and the Baire property, to appear.Google Scholar
  5. [K]
    K. Kunen,Set Theory, North-Holland, Amsterdam, 1980.MATHGoogle Scholar
  6. [O]
    A. Ostaszewski,On countably compact perfectly normal spaces, J. London Math. Soc.14 (1976), 505–516.MATHCrossRefMathSciNetGoogle Scholar
  7. [S]
    S. Shelah,Whitehead groups may not be free even assuming CH — Part II, Isr. J. Math.35 (1980), 257–285.MATHCrossRefGoogle Scholar
  8. [SW]
    J. Steprans and W. S. Watson,Homeomorphisms of manifolds with prescribed behaviour on large dense sets, Bull. London Math. Soc.19 (1987), 305–310.MATHCrossRefMathSciNetGoogle Scholar
  9. [W]
    W. Weiss,Versions of Martin’s axiom, inHandbook of Set-Theoretic Topology (K. Kunen and J. Vaughn, eds.), North-Holland, Amsterdam, 184, pp. 827–886.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1989

Authors and Affiliations

  • Stewart Baldwin
    • 1
  • Robert E. Beaudoin
    • 1
  1. 1.Department of MathematicsAuburn UniversityAuburnUSA

Personalised recommendations