A base-matrix lemma for sets of rationals modulo nowhere dense sets |
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Authors: | J?rg Brendle Diana Carolina Montoya |
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Institution: | 1. Graduate School of System Informatics, Kobe University, Kobe, Japan 2. Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia
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Abstract: | We study some properties of the quotient forcing notions ${Q_{tr(I)} = \wp(2^{< \omega})/tr(I)}$ and P I ?= B(2 ω )/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2 ω . We show that the remainder forcing R I =?Q tr(I)/P I is σ-closed in these cases. We also study the cardinal invariant of the continuum ${\mathfrak{h}_{\mathbb{Q}}}$ , the distributivity number of the quotient ${Dense(\mathbb{Q})/nwd}$ , in order to show that ${\wp(\mathbb{Q})/nwd}$ collapses ${\mathfrak{c}}$ to ${\mathfrak{h}_{\mathbb{Q}}}$ , thus answering a question addressed in Balcar et?al. (Fundamenta Mathematicae 183:59–80, 2004). |
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