The Hausdorff-Ershov hierarchy in Euclidean spaces

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Delta(ta)(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 Delta(B)(2) and Ershov's hierarchy in the class Delta(0)(2) of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Delta(ta)(2). This is based on suitable characterizations of the sets from Delta(ta)(2) which are obtained in a close analogy to those of the Delta(B)(2) sets as well as those of the Delta(0)(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 omega(2). Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented.