The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δta 2 is just large enough to include several types of pointsets in Euclidean spaces ℝ k which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class ΔB 2 and Ershov's hierarchy in the class Δ0 2 of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δta 2. This is based on suitable (...) characterizations of the sets from Δta 2 which are obtained in a close analogy to those of the ΔB 2 sets as well as those of the Δ0 2 sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω 2. Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented. (shrink)

We present some characterizations of the members of Δta2, that class of the topological arithmetical hierarchy which is just large enough to include several fundamental types of sets of points in Euclidean spaces ℝk. 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 Δta2. 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 Δta2. The presentation always first reports analogies between Hausdorff's difference hierarchy within the Borel class ΔB2 and Ershov's hierarchy within the class Δ02 of the arithmetical hierarchy; after that the counterparts for Δta2 are developed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)