Hierarchies in φ‐spaces and applications

We establish some results on the Borel and difference hierarchies in φ-spaces. Such spaces are the topological counterpart of the algebraic directed-complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non-collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω-ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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