Luzin Gaps

We isolate a class of Fσδ ideals on N that includes all analytic P-ideals and all Fσ ideals, and introduce ‘Luzin gaps’ in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author’s Rigidity Conjecture for quotients P(N)/I. We also prove that the ideals NWD(Q) and NULL(Q) have the Radon–Nikodym property, and (using OCA∞) an uniformization result for K-coherent families of continuous partial functions. One of the most fascinating facts about the Boolean algebra P(N)/Fin was discovered by Hausdorff in 1908. In [20], he constructed two families A and B of sets of integers such that (a) A ∩B is finite for all A ∈ A and all B ∈ B, (b) for every C ⊆ N either A \C is infinite for some A ∈ A or B ∩C is infinite for some B ∈ B, and (c) both and A and B have order-type equal to ω1 with respect to the inclusion modulo finite. Families A and B satisfying (a) are orthogonal, those satisfying both (a) and (b) form a gap in P(N)/Fin (or, they are not separated over Fin, the ideal of finite sets), and if they furthermore satisfy (c) they form an (ω1, ω1)-gap. If (a) and (b) hold and both A and B are linearly ordered by the inclusion modulo finite, a gap is linear. A gap (linear or not) is Hausdorff if both of its sides are σ-directed under the inclusion modulo finite. Another major advance in the study of gaps in P(N)/Fin was made by Kunen ([29]), who used a condition originally isolated by Luzin in [32] to prove that for every (ω1, ω1)-gap there is a ccc poset that freezes it. (A gap is indestructible, or frozen, if it remains a gap in every א1-preserving forcing extension.) Remarkably, a linear gap can be frozen by an א1-preserving forcing if and only if it contains an (ω1, ω1)-gap. Kunen used freezing to prove that P(N)/Fin need not be universal for linear orderings of size at most 2א0 even if Martin’s Axiom, MA, is assumed. The technique of freezing gaps has played an important role in a variety of subjects, from automatic continuity in Banach algebras ([6]) to the study of automorphisms of P(N)/Fin ([35]). Date: November 11, 2003. 1991 Mathematics Subject Classification. 03E50, 03E65, 06E05. Filename: luzin.tex. The author acknowledges support received from the National Science Foundation (USA) via grant DMS-0196153, PSC-CUNY grant #62785-00-31, the York University start-up grant, and