Computation and Logic in the Real World

Volume 4497 of the series Lecture Notes in Computer Science pp 83-97

Borel Complexity of Topological Operations on Computable Metric Spaces

  • Vasco BrattkaAffiliated withLaboratory of Foundational Aspects of Computer Science, Department of Mathematics & Applied Mathematics, University of Cape Town, Rondebosch 7701
  • , Guido GherardiAffiliated withDipartimento di Scienze Matematiche é, Informatiche Roberto Magari, University of Siena

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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.


Computable analysis effective descriptive set theory Borel measurability hyperspace topologies