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Descriptive Complexity on Non-Polish Spaces

Authors Antonin Callard, Mathieu Hoyrup

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Author Details

Antonin Callard
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Mathieu Hoyrup
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France


We want to thank Takayuki Kihara, Arno Pauly and Victor Selivanov for useful discussions on this topic.

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Antonin Callard and Mathieu Hoyrup. Descriptive Complexity on Non-Polish Spaces. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 8:1-8:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


Represented spaces are the spaces on which computations can be performed. We investigate the descriptive complexity of sets in represented spaces. We prove that the standard representation of a countably-based space preserves the effective descriptive complexity of sets. We prove that some results from descriptive set theory on Polish spaces extend to arbitrary countably-based spaces. We study the larger class of coPolish spaces, showing that their representation does not always preserve the complexity of sets, and we relate this mismatch with the sequential aspects of the space. We study in particular the space of polynomials.

Subject Classification

ACM Subject Classification
  • Theory of computation → Turing machines
  • Mathematics of computing → Point-set topology
  • Represented space
  • Computable analysis
  • Descriptive set theory
  • CoPolish spaces


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  1. Vasco Brattka. Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly, 51(1):19-44, 2005. URL:
  2. Matthew de Brecht. Quasi-Polish spaces. Ann. Pure Appl. Logic, 164(3):356-381, 2013. URL:
  3. Matthew de Brecht, Arno Pauly, and Matthias Schröder. Overt choice. CoRR, abs/1902.05926, 2019. URL:
  4. Matthew de Brecht, Matthias Schröder, and Victor Selivanov. Base-complexity classifications of qcb₀-spaces. Computability, 5(1):75-102, 2016. URL:
  5. Mathieu Hoyrup. Results in descriptive set theory on some represented spaces. Preprint, 2017. URL:
  6. Mathieu Hoyrup, Cristobal Rojas, Victor L. Selivanov, and Donald M. Stull. Computability on quasi-Polish spaces. In Michal Hospodár, Galina Jirásková, and Stavros Konstantinidis, editors, Descriptional Complexity of Formal Systems - 21st IFIP WG 1.02 International Conference, DCFS 2019, Košice, Slovakia, July 17-19, 2019, Proceedings, volume 11612 of Lecture Notes in Computer Science, pages 171-183. Springer, 2019. URL:
  7. Alexander S. Kechris. Classical Descriptive Set Theory. Springer, January 1995. Google Scholar
  8. Dag Normann. Chapter 8 - The continuous functionals. In Edward R. Griffor, editor, Handbook of Computability Theory, volume 140 of Studies in Logic and the Foundations of Mathematics, pages 251-275. Elsevier, 1999. URL:
  9. Arno Pauly. On the topological aspects of the theory of represented spaces. Computability, 5(2):159-180, 2015. URL:
  10. Arno Pauly and Matthew de Brecht. Descriptive set theory in the category of represented spaces. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015, pages 438-449. IEEE Computer Society, 2015. URL:
  11. Jean Saint Raymond. Preservation of the Borel class under countable-compact-covering mappings. Topology and its Applications, 154(8):1714-1725, 2007. URL:
  12. Matthias Schröder. Spaces allowing type-2 complexity theory revisited. Math. Log. Q., 50(4-5):443-459, 2004. URL:
  13. Victor L. Selivanov. Difference hierarchy in φ-spaces. Algebra and Logic, 43(4):238-248, July 2004. URL:
  14. Victor L. Selivanov. Total representations. Logical Methods in Computer Science, 9(2), 2013. URL:
  15. Victor L. Selivanov. Effective Wadge hierarchy in computable quasi-Polish spaces. Preprint, 2019. URL:
  16. Robert Vaught. Invariant sets in topology and logic. Fundamenta Mathematicae, 82(3):269-294, 1974. URL:
  17. Klaus Weihrauch. Computable Analysis. Springer, Berlin, 2000. Google Scholar
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