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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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