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Descriptive Complexity on Non-Polish Spaces II

Author Mathieu Hoyrup

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Mathieu Hoyrup
  • Université de Lorraine, CNRS, Inria, LORIA, Nancy, France


I want to thank the anonymous referees for their useful comments, and for suggesting the generalization of Corollary 2.4 to quasi-zero-dimensional spaces.

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Mathieu Hoyrup. Descriptive Complexity on Non-Polish Spaces II. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 132:1-132:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


This article is a study of descriptive complexity of subsets of represented spaces. Two competing measures of descriptive complexity are available. The first one is topological and measures how complex it is to obtain a set from open sets using boolean operations. The second one measures how complex it is to test membership in the set, and we call it symbolic complexity because it measures the complexity of the symbolic representation of the set. While topological and symbolic complexity are equivalent on countably-based spaces, they differ on more general spaces. Our investigation is aimed at explaining this difference and highly suggests that it is related to the well-known mismatch between topological and sequential aspects of topological spaces.

Subject Classification

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


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