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Sets with no uncountable Blackwell subsets. (English) Zbl 0637.28001

Assuming the continuum hypothesis, a set \(Y\subset {\mathbb{R}}\) is constructed such that none of its uncountable subsets is Blackwell. One may choose Y to be a Sierpiński or a Luzin set. The same construction works under the Martin’s axiom to get a set \(Y\subset {\mathbb{R}}\) of cardinality \({\mathfrak c}\) such that no subset of Y of cardinality \({\mathfrak c}\) is Blackwell. However, the sets of lower cardinality are Blackwell in this case.
Reviewer: P.Holicky

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
03E50 Continuum hypothesis and Martin’s axiom
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