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Logic and Analysis

, 1:235 | Cite as

Topometric spaces and perturbations of metric structures

  • Itaï Ben Yaacov
Article

Abstract

We develop the general theory of topometric spaces, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the ad hoc development in Ben Yaacov I and Usvyatsov A, Continuous first order logic and local stability. Trans Am Math Soc, in press), as well as of global \({\aleph_0}\)-stability. We conclude with a study of perturbation systems (see Ben Yaacov I, On perturbations of continuous structures, submitted) in the formalism of topometric spaces. In particular, we show how the abstract development applies to \({\aleph_0}\)-stability up to perturbation.

Keywords

Topometric spaces Continuous logic Type spaces Cantor-Bendixson rank Perturbation 

Mathematics Subject Classification (2000)

03C95 03C90 03C45 54H99 

References

  1. 1.
    Ben Yaacov I, Berenstein A (2008) On perturbations of hilbert spaces and probability algebras with a generic automorphism (in preparation)Google Scholar
  2. 2.
    Ben Yaacov I, Berenstein A, Henson CW, Usvyatsov A (2008) Model theory for metric structures. In: Model theory with applications to algebra and analysis, vol 2, Chatzidakis Z, Macpherson D, Pillay A, Wilkie A (eds) London Math Society Lecture Note Series, vol 350, pp 315–427Google Scholar
  3. 3.
    Ben Yaacov I (2005) Uncountable dense categoricity in cats. J Symbolic Log 70(3): 829–860zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ben Yaacov I Modular functionals and perturbations of Nakano spaces (submitted)Google Scholar
  5. 5.
    Ben Yaacov I (2008) On perturbations of continuous structures (submitted)Google Scholar
  6. 6.
    Ben Yaacov I, Usvyatsov A (2007) On d-finiteness in continuous structures. Fundam Math 194: 67–88zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ben Yaacov I, Usvyatsov A (2008) Continuous first order logic and local stability. Trans Am Math Soc (in press)Google Scholar
  8. 8.
    Iovino J (1999) Stable Banach spaces and Banach space structures, I and II. Models, algebras, and proofs (Bogotá, 1995). Lecture Notes in Pure and Applied Mathematics, vol 203. Dekker, New York, pp 77–117Google Scholar
  9. 9.
    Keisler HJ (1987) Choosing elements in a saturated model. In: Classification theory (Chicago, IL, 1985). Lecture Notes in Mathematics, vol 1292. Springer, Berlin, pp 165–181Google Scholar
  10. 10.
    Keisler HJ (1987) Measures and forking. Ann Pure Appl Log 34(2): 119–169zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Newelski L (2003) The diameter of a Lascar strong type. Fundam Math 176(2): 157–170zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Itaï Ben Yaacov
    • 1
  1. 1.Université de Lyon, Université Lyon 1Institut Camille Jordan, CNRS UMR 5208Villeurbanne CedexFrance

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