Computability
Theoretical Computer Science - Special issue on real numbers and computers
Computability on subsets of Euclidean space I: closed and compact subsets
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computable analysis: an introduction
Computable analysis: an introduction
Theoretical Computer Science
Random Numbers and an Incomplete Immune Recursive Set
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
Computable Exchangeable Sequences Have Computable de Finetti Measures
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
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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.