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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.
MSC:
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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