Characterizations of the class Δ_2~(ta) over Euclidean spaces

摘要

We present some characterizations of the members of Δ_2~(ta), that class of the topological arithmetical hierarchy which is just large enough to include several fundamental types of sets of points in Euclidean spaces R~k. The limit characterization serves as a basic tool in further investigations. The characterization by effective difference chains of effectively exhaustible sets yields only a hierarchy within a subfield of Δ_2~(ta). Effective difference chains of transfinite (but constructive) order types, consisting of complements of effectively exhaustible sets, as well as another closely related concept, yield a rich hierarchy within the whole class Δ_2~(ta). The presentation always first reports analogies between Hausdorff's difference hierarchy within the Borel class B2 and Ershov's hierarchy within the class 02 of the arithmetical hierarchy; after that the counterparts for Δ_2~(ta) are developed.