Slices: Functions for abstract real analysis

  • Robert O. Robson
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1420)


Let F be a real closed field and let Open image in new window . If Open image in new window in an open constructible set, an abstract function on V is a section of the natural projection π : Open image in new window over V. In this paper we undertake a study of abstract functions in general, extending the notions of boundedness and compatibility from the work of N. Schwartz, H. Delfs, and others. We then introduce of a nice a class of abstract functions, called slices, whose values are determined by semialgebraic approximations. Familiar transcendental functions provide examples over R. We give criteria for extending Open image in new window -valued functions on Fn to slices for arbitrary F. This paper is in its final form and no similar paper has been submitted elsewhere.


Open Neighborhood Abstract Function Real Analytic Function Real Spectrum Quotient Field 
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. [1]
    E. Becker, On the real spectrum of a ring and its application to semialgebraic geometry, Bull. Amer. Math. Soc. 15 (1986), 19–60.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. Bochnak, M. Coste, M.-F. Roy, Géometrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Serie 12 Springer-Verlag, New York, Berlin, Heidelberg.Google Scholar
  3. [3]
    G. Brumfiel, The real spectrum of an ideal and KO-theory exact sequences, K-theory 1 (1987), 211–235.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    H. Delfs, The homotopy axiom in semialgebraic cohomology, J. reine u. angew. Math. 355 (1985) 108–128.MathSciNetzbMATHGoogle Scholar
  5. [5]
    H. Delfs and M. Knebusch, Separation, retractions, and homotopy extension in semialgebraic spaces, Pacific J. of Math. 114 (1984), 47–71.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Prieß-Crampe, Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen, Ergebnisse der Mathematik und ihrer Grenzgebiete 98, Springer-Verlag, New York, Berlin, Heidelberg.Google Scholar
  7. [7]
    N. Schwartz, Real Closed Rings, in Agebra and Order: Proc. 1 st Int. Symp. Ordered Algerbraic Structures Luminy-Marseille 1984. S. Wolfenstein (ed.) Heldermann Verlag, Berlin (1986) 175–194.Google Scholar
  8. [8]
    N. Schwartz, The basic theory of real closed spaces, Regensburger Mathematische Schriften 15 (1987). (Available from NWF I-Mathematik, Universität, Postfach 397, D-8400 Regensburg, West Germany).Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Robert O. Robson
    • 1
  1. 1.Department of MathematicsOregon State UniversityCorvallis

Personalised recommendations