Skip to main content
SpringerLink
Log in
Book cover

Set Theory pp 345–400Cite as

Local Connectedness and Distance Functions

  • Chapter

Part of the book series: Trends in Mathematics ((TM))

Abstract

Local connectedness functions for (κ, 1)-simplified morasses, localisations of the coupling function c studied in [M96, §1], are defined and their elementary properties discussed. Several different useful canonical ways of arriving at the functions are examined. This analysis is then used to give explicit formulae for generalisations of the local distance functions introduced with a recursive definition in [K00], leading to simple proofs of the principal properties of those functions. It is then also extended to the properties of local connectedness functions in the context of \( \kappa {\text{ - }}\mathbb{M} \)-proper forcing for successor κ. The functions are shown to enjoy substantial strengthenings of the properties (particularly the Δ-properties) hitherto proved for both the function c and for Todorcevic’s ρ-functions in the special case κ = ω1. A couple of examples of the use of local connectedness functions in consort with \( \kappa {\text{ - }}\mathbb{M} \)-proper forcing are then given.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. U. Abraham and S. Shelah, Forcing closed unbounded sets, Journal of Symbolic Logic, 48, (1983), pp. 643–657.

    Article  MATH  MathSciNet  Google Scholar 

  2. U. Abraham and R. Shore, Initial segements of the degrees of size ℵ1, Israel Journal of Mathematics, 53, (1986), pp. 1–51.

    MATH  MathSciNet  Google Scholar 

  3. J. Buamgartner, Almost-disjoint sets, the dense set problem and the partition calculus, Annals of Mathematical Logic, 10, (1976), pp. 401–439.

    Article  Google Scholar 

  4. J. Baumgartner and S. Shelah, Remarks on superatomic Boolean algebras, Annals of Pure and Applied Logic, 33, (1987), pp. 109–129.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Erdős and A. Hajnal, Unsolved and solved problems in set theory, Proceedings of the Symposia in Pure Mathematics, 25, (1974), pp. 267–287.

    Google Scholar 

  6. W. Gowers, The two cultures of mathematics, in Mathematics: frontiers and perspectives, ed. V.I. Arnold, M.A. Atiyah, P. Lax and B. Mazur, American Mathematical Society, Providence, Rhode Island, U.S.A., 2000, pp. 65–78.

    Google Scholar 

  7. P. Koepke and J.C. Martinez, Superatomic Boolean algebras constructed from morasses, Journal of Symbolic Logic, 60, (1995), pp. 940–951.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Komjath, A consistency results concerning set mappings, Acta Mathematica Hungarica, 64, (1994), pp. 93–99.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Koszmider, Semimorasses and nonreflection a singular cardinals, Annals of Pure and Applied Logic, 72, (1995), pp. 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Koszmider, On the existence of strong chains in P(ω 1)/Fin, Journal of Symbolic Logic, 63, (1998), pp. 1055–1062.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Koszmider, Models as side conditions, in Set theory, ed. C.A. Di Prisco, J. Larson, J. Bagaria and A.R.D. Mathias, Kluwer, Amsterdam, Holland, (1998), pp. 99–107.

    Google Scholar 

  12. P. Koszmider, On strong chains of uncountable functions, Israel Journal of Mathematics, 118, (2000), pp. 289–315.

    MATH  MathSciNet  Google Scholar 

  13. C.J.G. Morgan, Morasses, square and forcing axioms, Annals of Pure and Applied Logic, 80, (1996), pp. 139–163.

    Article  MATH  MathSciNet  Google Scholar 

  14. C.J.G. Morgan, A gap cohomology group, II: the Abraham-Shore technique, Journal of Symbolic Logic, to appear.

    Google Scholar 

  15. C.J.G. Morgan, pcf-Structures, I: characterisation, preprint, (2001).

    Google Scholar 

  16. C.J.G. Morgan, The uses and abuses of historicised forcing, handwritten notes, (1992), revised preprint, (2001).

    Google Scholar 

  17. C.J.G. Morgan, pcf-Structures, II: the consistency of pcf structures on ω2, preprint, (2001).

    Google Scholar 

  18. C.J.G. Morgan, A few gentle stretching exercises, CRM preprint. bf 554, (2003).

    Google Scholar 

  19. C.J.G. Morgan, Adding club subsets of ω2 using conditions with finite working parts, CRM preprint, (2004).

    Google Scholar 

  20. C.J.G. Morgan, On the Velickovic Δ-property for the stepping up functions c and ρ, submitted.

    Google Scholar 

  21. C.J.G. Morgan, pcf-structures, III: structures below ω3, in preparation.

    Google Scholar 

  22. M. Rabus, An ω2-minimal Boolean algebra, Proceedings of The American Mathematical Society, 348, (1996), pp. 3235–3244.

    MATH  MathSciNet  Google Scholar 

  23. J. Roitman, Height and width of superatomic Boolean algebras, Proceedings of the American Mathematical Society, 94, (1985), pp. 9–14.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Shelah, Not collapsing cardinals ≤ κ in (< κ)-support iterations, preprint.

    Google Scholar 

  25. S. Shelah, C. Laflamme and B. Hart, Models with second-order properties, V, a general principle, Annals of Pure and Applied Logic, 64, (1993), pp. 169–194.

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Shelah, L. Stanley, A theorem and some consistency results in partition calculus, Annals of Pure and Applied Logic, 36, (1987), pp. 119–152.

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Todorčević, Directed sets and cofinal types, Transactions of the American Mathematical Society, 290, (1985), pp. 711–723.

    Article  MathSciNet  Google Scholar 

  28. S. Todorčević, Partitioning pairs of countable ordinals, Acta Mathematica, 159, (1987), pp. 261–294.

    Article  MathSciNet  Google Scholar 

  29. S. Todorčević, Partition problems in topology, Contemporary Mathematics, 84, American Mathematical Society, Providence, Rhode Island, U.S.A., 1989.

    Google Scholar 

  30. D. Velleman, Simplified morasses, Journal of Symbolic Logic, 49, (1984), pp. 257–271.

    Article  MATH  MathSciNet  Google Scholar 

  31. B. Veličković, Forcing axioms and stationary sets, Advances in Mathematics, 94, (1992), pp. 256–284

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, Great Britain

    Charles Morgan

Authors
  1. Charles Morgan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departament de Lògica, Història I Filosofia de la Ciència, Universitat de Barcelona, Baldiri Reixac s/n, 08028, Barcelona, Spain

    Joan Bagaria

  2. Department of Mathematics, University of Toronto, Toronto, M5S 2E4, Canada

    Stevo Todorcevic

  3. C.N.R.S., UMR 7056, Université Paris 7, 2, Place Jussieu Case 7012, 75251, Paris Cedex 05, France

    Stevo Todorcevic

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Morgan, C. (2006). Local Connectedness and Distance Functions. In: Bagaria, J., Todorcevic, S. (eds) Set Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7692-9_14

Download citation

Publish with us

Policies and ethics

Search

Navigation