## 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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### References:

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