Singular Coverings and Non-Uniform Notions of Closed Set Computability

The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskĭı, Tsĕıtin, Kreisel, and Lacombe assert the existence of non-empty co-r.e. closed sets devoid of computable points: sets which are even ‘large’ in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary : every non-empty co-r.e. closed real set without computable points has continuum cardinality. This initiates a comparison of different notions of computability for closed real subsets non-uniformly like, e.g., for sets of fixed cardinality or sets containing a (not necessarily effectively findable) computable point. By relativization we obtain a bounded recursive rational sequence of which every accumulation point is not even computable with support of a Halting oracle. Finally the question is treated whether compact sets have co-r.e. closed connected components; and every star-shaped co-r.e. closed set is asserted to contain a computable point.