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On the equivalence between CH and the existence of certain \({\mathcal I}\)-Luzin subsets of \(\mathbb{R}\). (English) Zbl 1050.03033
Summary: We extend Rothberger’s theorem (on the equivalence between CH and the existence of Luzin and Sierpiński-sets having power \({\mathfrak c})\) and certain paradoxical constructions due to Erdős. More precisely, by employing a suitable \(\sigma\)-ideal associated to the \((\alpha, \beta)\)-games introduced by Schmidt, we prove that the continuum hypothesis holds if and only if there exist subgroups of \((\mathbb{R}, +)\) having power \({\mathfrak c}\) and intersecting every “absolutely losing” (respectively, every meager and null) set in at most countably many points.
03E15 Descriptive set theory
03E50 Continuum hypothesis and Martin’s axiom
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
91A44 Games involving topology, set theory, or logic
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