کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4661847 | 1633464 | 2014 | 24 صفحه PDF | دانلود رایگان |
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):(1)Every product of a concentrated space with a Hurewicz S1(Γ,O)S1(Γ,O) space satisfies S1(Γ,O)S1(Γ,O). 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.(2)Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger.(3)Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits–Nagy.(4)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 add(N)add(N), the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy (u
Journal: Annals of Pure and Applied Logic - Volume 165, Issue 5, May 2014, Pages 1034–1057