Some interesting problems

A ≤W B. (This refers to Wadge reducible.) Answer: The first question was answered by Hjorth [83] who showed that it is independent. 1.2 A subset A ⊂ ω ω is compactly-Γ iff for every compact K ⊂ ω ω we have that A ∩ K is in Γ. Is it consistent relative to ZFC that compactly-Σ 1 1 implies Σ 1 1? (see Miller-Kunen [111], Becker [11]) 1.3 (Miller [111]) Does ∆ 1 1 = compactly- ∆ 1 1 imply Σ 1 1 = compactly-Σ 1 1? 1.4 (Prikry see [62]) Can L ∩ ω ω be a nontrivial Σ 1 1 set? Can there be a nontrivial perfect set of constructible reals? Answer: No, for first question Velickovic-Woodin [192]. question Groszek-Slaman [71]. See also Gitik [67]