Selective covering properties of product spaces

Annals of Pure and Applied Logic 165 (5):1034-1057 (2014)
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Abstract

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals.Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are : Every product of a concentrated space with a Hurewicz S1S1 space satisfies S1S1. On the other hand, assuming the Continuum Hypothesis, for each Sierpiński set S there is a Luzin set L such that L×SL×S can be mapped onto the real line by a Borel function. Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits–Nagy. Assuming d=ℵ1d=ℵ1, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers.A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than addadd, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy >ℵ1.Our results improve upon and unify a number of results, established earlier by many authors

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Citations of this work

Products of Hurewicz Spaces in the Laver Model.Dušan Repovš & Lyubomyr Zdomskyy - 2017 - Bulletin of Symbolic Logic 23 (3):324-333.
Products of Menger Spaces: A Combinatorial Approach.Piotr Szewczak & Boaz Tsaban - 2017 - Annals of Pure and Applied Logic 168 (1):1-18.

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References found in this work

Borel Sets and Ramsey's Theorem.Fred Galvin & Karel Prikry - 1973 - Journal of Symbolic Logic 38 (2):193-198.
Near Coherence of Filters. I. Cofinal Equivalence of Models of Arithmetic.Andreas Blass - 1986 - Notre Dame Journal of Formal Logic 27 (4):579-591.
On the Cofinality of Ultrapowers.Andreas Blass & Heike Mildenberger - 1999 - Journal of Symbolic Logic 64 (2):727-736.

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