Random Numbers and an Incomplete Immune Recursive Set

Generalizing the notion of a recursively enumerable (r.e.) set to sets of real numbers 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).