Borel Complexity of Topological Operations on Computable Metric Spaces

  • Vasco Brattka
  • Guido Gherardi
Conference paper

DOI: 10.1007/978-3-540-73001-9_9

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)
Cite this paper as:
Brattka V., Gherardi G. (2007) Borel Complexity of Topological Operations on Computable Metric Spaces. In: Cooper S.B., Löwe B., Sorbi A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg

Abstract

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.

Keywords

Computable analysis effective descriptive set theory Borel measurability hyperspace topologies 

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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

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