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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