Skip to main content
SpringerLink
Log in

Continuous reducibility and dimension of metric spaces

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

If (X, d) is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space (X, d) of positive dimension, there are uncountably many Borel subsets of (X, d) that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \((X,\tau )\) is called the Wadge quasi-order for \((X,\tau )\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \((X,\tau )\), is a well-quasiorder if and only if \((X,\tau )\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andretta, A., Martin, D.A.: Borel-Wadge degrees. Fund. Math. 177(2), 175–192 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andretta, A.: More on Wadge determinacy. Ann. Pure Appl. Logic 144(1–3), 2–32 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andretta, A.: The SLO principle and the Wadge hierarchy. In: Bold, S., Löwe, B., Räsch, T., van Benthem, J. (eds.) Foundations of the Formal Sciences V, Volume 11 of Studies in Logic, pp. 1–38. College Publications, London (2007)

  4. Carroy, R.: A quasi-order on continuous functions. J. Symb. Logic 78(2), 633–648 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cook, H.: Continua which admit only the identity mapping onto non-degenerate subcontinua. Fund. Math. 60, 241–249 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carroy, R., Pequignot, Y.: From well to better, the space of ideals. Fund. Math. 227(3), 247–270 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dijkstra, J.J., van Mill, J.: Characterizing complete Erdős space. Can. J. Math. 61(1), 124–140 (2009)

    Article  MATH  Google Scholar 

  8. Elekes, M.: Linearly ordered families of Baire 1 functions. Real Anal. Exch. 27(1), 49–63 (2001–2002)

  9. Engelking, R.: General Topology, Volume 6 of Sigma series in pure mathematics, second edition. Springer, Berlin (1989). Translated from the Polish by the author

  10. Hurewicz, W., Wallman, H.: Dimension Theory. Princeton mathematical series, vol. 4. Princeton University Press, Princeton (1941)

  11. Jech, T.: Set Theory. Springer monographs in mathematics. Springer, Berlin (2003). The third millennium edition, revised and expanded

  12. Kechris, A.S.: Classical Descriptive Set Theory. Graduate texts in mathematics, vol. 156. Springer, New York (1995)

  13. Louveau, A.: Some results in the Wadge hierarchy of Borel sets. In: Kechris, A.S., Löwe, B., Steel, J.R. (eds.) Wadge Degrees and Projective Ordinals. The Cabal Seminar. Volume II, Volume 37 of Lecture Notes Logic, pp. 47–74. Association of Symbolic Logic, La Jolla (2012)

  14. Motto Ros, L.: Borel-amenable reducibilities for sets of reals. J. Symb. Logic 74(1), 27–49 (2009)

  15. Motto Ros, L.: Baire reductions and good Borel reducibilities. J. Symb. Logic 75(1), 323–345 (2010)

  16. Motto Ros, L.: Beyond Borel-amenability: scales and superamenable reducibilities. Ann. Pure Appl. Logic 161(7), 829–836 (2010)

  17. Motto Ros, L.: Bad Wadge-like reducibilities on the Baire space. Fund. Math. 224(1), 67–95 (2014)

  18. Motto Ros, L., Schlicht, P.: Lipschitz and uniformly continuous reducibilities on ultrametric polish spaces. In: Logic, computation, hierarchies. Ontos Mathematical Logic, vol. 4, pp. 213–258. De Gruyter, Berlin (2014)

  19. Motto Ros, L., Schlicht, P., Selivanov, V.: Wadge-like reducibilities on arbitrary quasi-Polish spaces. Math. Struct. Comput. Sci. 25(8), 1705–1754 (2015)

  20. Pequignot, Y.: A Wadge hierarchy for second countable spaces. Arch. Math. Logic 54(5–6), 659–683 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wadge, W.W.: Early investigations of the degrees of Borel sets. In: Kechris, A.S., Löwe, B., Steel, J.R. (eds.) Wadge Degrees and Projective Ordinals. The Cabal Seminar. Volume II, Volume 37 of Lecture Notes Logic, pp. 166–195. Association of Symbolic Logic, La Jolla (2012)

  22. Woodin, W.H.: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Volume 1 of de Gruyter Series in Logic and its Applications. Walter de Gruyter GmbH & Co. KG, Berlin, revised edition (2010)

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Philipp Schlicht

Authors
  1. Philipp Schlicht
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philipp Schlicht.

Additional information

This paper supersedes an earlier preprint, in which the main result of this paper was proved for Polish spaces by an unrelated proof.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schlicht, P. Continuous reducibility and dimension of metric spaces. Arch. Math. Logic 57, 329–359 (2018). https://doi.org/10.1007/s00153-017-0571-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-017-0571-6

Keywords

Mathematics Subject Classification

Search

Navigation