This is an extended abstract presenting new results on the topological complexity of omega-powers (which are included in a paper Classical and effective descriptive complexities of omega-powers available from arXiv:0708.4176) and reflecting also some open questions which were discussed during the Dagstuhl seminar on Topological and Game-Theoretic Aspects of Infinite Computations 29.06.08 - 04.07.08.
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