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Journal of Logic and Computation Advance Access originally published online on August 6, 2008
Journal of Logic and Computation 2009 19(1):45-76; doi:10.1093/logcom/exn027
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© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

This article appears in the following Journal of Logic and Computation issue: Special Issue: Logic and Computation in the Real World: CiE 2007 [View the issue table of contents]

Original Articles

Borel Complexity of Topological Operations on Computable Metric Spaces

Vasco Brattka *

Laboratory of Foundational Aspects of Computer Science, Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa.
E-mail: vasco.brattka{at}uct.ac.za

Guido Gherardi

Dipartimento di Filosofia, University of Bologna, Italy.
E-mail: guido.gherardi{at}unibo.it

Received 22 February 2008.


  Abstract

We study the Borel complexity of topological operations on closed subsets of computable metric spaces. The investigated operations include set theoretic operations such 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 recursively enumerable (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; effective Borel complexity; theory of representations; Fell topology; Effros Borel structure

*This work has been partially supported by the National Research Foundation (NRF).


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