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E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non\u2010collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space<jats:italic>P\u03c9<\/jats:italic>. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the<jats:italic>\u03c9<\/jats:italic>\u2010ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (\u00a9 2004 WILEY\u2010VCH Verlag GmbH &amp; Co. 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