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
Article

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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