Luzin gaps

Abstract

We isolate a class of $F_{\sigma \delta }$ ideals on $\mathbb {N}$ that includes all analytic P-ideals and all $F_\sigma$ 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 $\mathcal {P}(\mathbb {N})/\mathcal {I}$. We also prove that the ideals $\operatorname {NWD}(\mathbb {Q})$ and $\operatorname {NULL}(\mathbb {Q})$ have the Radon–Nikodým property, and (using OCA$_\infty$) a uniformization result for $\mathcal {K}$-coherent families of continuous partial functions.