Borel Complexity of Topological Operations on Computable Metric Spaces

  • Vasco Brattka
  • Guido Gherardi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)


We study the Borel complexity of topological operations on closed subsets of computable metric spaces. The investigated operations include set theoretic operations as union and intersection, but also typical topological operations such as the closure of the complement, the closure of the interior, the boundary and the derivative of a set. These operations are studied with respect to different computability structures on the hyperspace of closed subsets. These structures include positive or negative information on the represented closed subsets. Topologically, they correspond to the lower or upper Fell topology, respectively, and the induced computability concepts generalize the classical notions of r.e. or co-r.e. subsets, respectively. The operations are classified with respect to effective measurability in the Borel hierarchy and it turns out that most operations can be located in the first three levels of the hierarchy, or they are not even Borel measurable at all. In some cases the effective Borel measurability depends on further properties of the underlying metric spaces, such as effective local compactness and effective local connectedness.


Computable analysis effective descriptive set theory Borel measurability hyperspace topologies 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theoretical Computer Science 305, 43–76 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kechris, A.S.: Classical Descriptive Set Theory. Volume 156 of Graduate Texts in Mathematics. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. 4.
    Moschovakis, Y.N.: Descriptive Set Theory. Volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1980)Google Scholar
  5. 5.
    Brattka, V.: Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly 51, 19–44 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kuratowski, K.: Topology, vol. 1. Academic Press, London (1966)zbMATHGoogle Scholar
  7. 7.
    Kuratowski, K.: Topology, vol. 2. Academic Press, London (1968)Google Scholar
  8. 8.
    Christensen, J.P.R.: On some properties of Effros Borel structure on spaces of closed subsets. Mathematische Annalen 195, 17–23 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Christensen, J.P.R.: Necessary and sufficient conditions for the measurability of certain sets of closed subsets. Mathematische Annalen 200, 189–193 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Christensen, J.P.R.: Topology and Borel Structure. North-Holland, Amsterdam (1974)zbMATHGoogle Scholar
  11. 11.
    Brattka, V.: Computable invariance. Theoretical Computer Science 210, 3–20 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gherardi, G.: Effective Borel degrees of some topological functions. Mathematical Logic Quarterly 52, 625–642 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Weihrauch, K.: Computability. Volume 9 of EATCS Monographs on Theoretical Computer Science. Springer, Heidelberg (1987)Google Scholar
  14. 14.
    Kreitz, C., Weihrauch, K.: Theory of representations. Theoretical Computer Science 38, 35–53 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schröder, M.: Extended admissibility. Theoretical Computer Science 284, 519–538 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Brattka, V., Weihrauch, K.: Computability on subsets of Euclidean space I: Closed and compact subsets. Theoretical Computer Science 219, 65–93 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Brattka, V.: Random numbers and an incomplete immune recursive set. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 8–13. Springer, Heidelberg (2002)Google Scholar
  18. 18.
    Holá, L., Pelant, J., Zsilinszky, L.: Developable hyperspaces are metrizable. Applied General Topology 4, 351–360 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Raymond, J.S.: La structure borélienne d’Effros est-elle standard? Fundamenta Mathematicae 100, 201–210 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Effros, E.G.: Convergence of closed subsets in a topological space. Proceedings of the American Mathematical Society 16, 929–931 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Brattka, V.: Plottable real number functions. In: Daumas, M., et al. (eds.) RNC’5 Real Numbers and Computers, INRIA, Institut National de Recherche en Informatique et en Automatique 13–30 Lyon, September 3–5, 2003 (2003)Google Scholar
  22. 22.
    Cenzer, D., Mauldin, R.D.: On the Borel class of the derived set operator. Bull. Soc. Math. France 110, 357–380 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cenzer, D., Mauldin, R.D.: On the Borel class of the derived set operator: II. Bull. Soc. Math. France 111, 367–372 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kuratowski, K.: Some remarks on the relation of classical set-valued mappings to the Baire classification. Colloquium Mathematicum 42, 273–277 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Vasco Brattka
    • 1
  • Guido Gherardi
    • 2
  1. 1.Laboratory of Foundational Aspects of Computer Science, Department of Mathematics & Applied Mathematics, University of Cape Town, Rondebosch 7701South Africa
  2. 2.Dipartimento di Scienze Matematiche é, Informatiche Roberto Magari, University of SienaItaly

Personalised recommendations