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Property C'', strong measure zero sets and subsets of the plane

Janusz Pawlikowski — 1997

Fundamenta Mathematicae

Let X be a set of reals. We show that  • X has property C" of Rothberger iff for all closed F ⊆ ℝ × ℝ with vertical sections F x (x ∈ X) null, x X F x is null;  • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections F x (x ∈ ℝ) null, x X F x is null.

Strongly meager sets and subsets of the plane

Janusz Pawlikowski — 1998

Fundamenta Mathematicae

Let X 2 w . Consider the class of all Borel F X × 2 w with null vertical sections F x , x ∈ X. We show that if for all such F and all null Z ⊆ X, x Z F x is null, then for all such F, x X F x 2 w . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

Selivanovski hard sets are hard

Janusz Pawlikowski — 2015

Fundamenta Mathematicae

Let H Z 2 ω . For n ≥ 2, we prove that if Selivanovski measurable functions from 2 ω to Z give as preimages of H all Σₙ¹ subsets of 2 ω , then so do continuous injections.

Parametrized Cichoń's diagram and small sets

Janusz PawlikowskiIreneusz Recław — 1995

Fundamenta Mathematicae

We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of w w × 2 w and continuous functions e , f : w w w w such that  • N is G δ and N x : x w w , the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of 2 w ;  • M is F σ and M x : x w w is a basis for the ideal of meager subsets of 2 w ;  • x , y N e ( x ) N y M x M f ( y ) . From this we derive that for a separable metric space X,  •if for all Borel (resp. G δ ) sets B X × 2 w with all...

Covering Property Axiom C P A c u b e and its consequences

Krzysztof CiesielskiJanusz Pawlikowski — 2003

Fundamenta Mathematicae

We formulate a Covering Property Axiom C P A c u b e , which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence f n < ω of Borel functions there are sequences:...

Uncountable γ-sets under axiom C P A c u b e g a m e

Krzysztof CiesielskiAndrés MillánJanusz Pawlikowski — 2003

Fundamenta Mathematicae

We formulate a Covering Property Axiom C P A c u b e g a m e , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < , since every γ-set is universally null, while C P A c u b e g a m e implies that every universally null has cardinality less than = ω₂. We also show that C P A c u b e g a m e implies the existence of a partition of ℝ into ω₁ null compact sets....

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