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On analytic filters and prefilters. (English) Zbl 0705.03027

Summary: We show that every analytic filter is generated by a \(\Pi^ 0_ 2\)- prefilter, every \(\Sigma^ 0_ 2\) filter is generated by a \(\Pi^ 0_ 1\) prefilter, and if \(P\subseteq {\mathcal P}(\omega)\) is a \(\Sigma^ 0_ 2\) prefilter then the filter generated by it is also \(\Sigma^ 0_ 2\). The last result is unique for the Borel classes, as there is a \(\Pi^ 0_ 2\)-complete prefilter P such that the filter generated by it is \(\Sigma^ l_ 1\)-complete. Also, no complete coanalytic filter is generated by an analytic prefilter. The proofs use König’s infinity lemma, a normal form theorem for monotone analytic sets, and Wadge reductions.

MSC:

03E15 Descriptive set theory
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