Characterizations of the class Δta2 over Euclidean spaces
✍ Scribed by Armin Hemmerling
- Publisher
- John Wiley and Sons
- Year
- 2004
- Tongue
- English
- Weight
- 240 KB
- Volume
- 50
- Category
- Article
- ISSN
- 0044-3050
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✦ Synopsis
Abstract
We present some characterizations of the members of Δ^ta^~2~, 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^ta^~2~. 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^ta^~2~. The presentation always first reports analogies between Hausdorff's difference hierarchy within the Borel class Δ^B^~2~ and Ershov's hierarchy within the class Δ^0^~2~ of the arithmetical hierarchy; after that the counterparts for Δ^ta^~2~ are developed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)