On the equivalence between CH and the existence of certain \({\mathcal I}\)-Luzin subsets of \(\mathbb{R}\). (English) Zbl 1050.03033

Zoli, Enrico

Georgian Math. J. 11, No. 1, 195-202 (2004).

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

Keywords:

continuum hypothesis; Schmidt games; \({\mathcal I}\)-Luzin sets; \(\sigma\)-ideals; vector subspaces of \(\mathbb{R}\) over the rationals