Abstract

We show that every analytic filter is generated by a Π 0 2 prefilter, every Σ 0 2 filter is generated by a Π 0 1 prefilter, and if $P \subseteq \mathscr{P}(\omega)$ is a Σ 0 2 prefilter then the filter generated by it is also Σ 0 2 . The last result is unique for the Borel classes, as there is a Π 0 2 -complete prefilter P such that the filter generated by it is Σ 1 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