2013/03/14 by Arnold W. Miller, Boaz Tsaban, Lyubomyr Zdomskyy · 2 citations
Mathematics · #Additive function #Advanced Banach Space Theory #Advanced Topology and Set Theory #Combinatorics #Computer science #Discrete mathematics #Lebesgue integration #Limits and Structures in Graph Theory #Mathematics #Modulo #Null set #Power set #Product (mathematics) #Product measure #Product topology #Set (abstract data type) #Set function #Space (punctuation) #Trichotomy (philosophy) #math.GN #math.LO
paper · pdf · doi:10.1016/j.apal.2014.01.001
published as Annals of Pure and Applied Logic 165 (2014), 1034-1057 · Submitted for publication
arxiv created 2013/03/14 · openalex publication_date 2014/01/22 · arxiv updated 2014/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 (definitions provided in the paper): \be \item Every product of a concentrated space with a Hurewicz \sone(\Ga,\Op) space satisfies \sone(\Ga,\Op). On the other hand, assuming \CH, for each Sierpiński set S there is a Luzin set L such that L\x S can be mapped onto the real line by a Borel function. \item Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. \item Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits--Nagy. \item Assuming \fd=ℵ1, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers. \ee A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than \add(\cN), the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy with \add(\cN)<\cov(\cM). Our results improve upon and unify a number of results, established earlier by many authors.