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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Luzin gaps


Author: Ilijas Farah
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2197-2239
MSC (2000): Primary 03E50, 03E65, 06E05
DOI: https://doi.org/10.1090/S0002-9947-04-03565-2
Published electronically: February 2, 2004
MathSciNet review: 2048515
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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.


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Additional Information

Ilijas Farah
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3 – and – Matematicki Institut, Kneza Mihaila 35, Belgrade
Email: ifarah@mathstat.yorku.ca

DOI: https://doi.org/10.1090/S0002-9947-04-03565-2
Received by editor(s): October 15, 2001
Published electronically: February 2, 2004
Additional Notes: 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 the NSERC (Canada)
Article copyright: © Copyright 2004 American Mathematical Society