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 Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe have asserted 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.This leads us to investigate for various classes of computable real subsets whether they always contain a (not necessarily effectively findable) computable point.