Complexity theory of real functions
Complexity theory of real functions
Computability on computable metric spaces
Theoretical Computer Science
Computability on subsets of Euclidean space I: closed and compact subsets
Theoretical Computer Science - Special issue on computability and complexity in analysis
Effective properties of sets and functions in metric spaces with computability structure
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computable analysis: an introduction
Computable analysis: an introduction
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
Borel Complexity of Topological Operations on Computable Metric Spaces
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
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Generalizing the notion of a recursively enumerable (r.e.) set to sets of realn umbers and other metric spaces is an important topic in computable analysis (which is the Turing machine based theory of computable real number functions). A closed subset of a computable metric space is called r.e. closed, if all open rational balls which intersect the set can be effectively enumerated and it is called effectively separable, if it contains a dense computable sequence. Both notions are closely related and in case of Euclidean space (and complete computable metric spaces in general) they actually coincide. Especially, both notions are generalizations of the classical notion of an r.e. subset of natural numbers. However, in case of incomplete metric spaces these notions are distinct. We use the immune set of random natural numbers to construct a recursive immune "tree" which shows that there exists an r.e. closed subset of some incomplete subspace of Cantor space which is not effectively separable. Finally, we transfer this example to the incomplete space of rational numbers (considered as a subspace of Euclidean space).