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.
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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].
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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.
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We show that a set of reals is undetermined in Galvin's point-open game iff it is uncountable and has property C", which answers a question of Gruenhage.
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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 vertical sections null, $∪_{x ∈ X}B_x$ is null, then for all Borel (resp. $F_σ$) sets $B ⊆ X×2^w$ with all vertical sections meager, $∪_{x ∈ X}B_x$ is meager; •if there exists a Borel (resp. a "nice" $G_δ$) set $B ⊆ X×2^w$ such that ${B_x:x ∈ X}$ is a basis for measure zero sets, then there exists a Borel (resp. $F_σ$) set $B ⊆ X×2^w$ such that ${B_x:x ∈ X}$ is a basis for meager sets
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We formulate a Covering Property Axiom $CPA_{cube}$, 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. (c) cof(𝓝) = ω₁ < 𝔠, i.e., the cofinality of the measure ideal 𝓝 is ω₁. (d) For every uniformly bounded sequence $⟨fₙ ∈ ℝ^{ℝ}⟩_{n<ω}$ of Borel functions there are sequences: $⟨P_ξ ⊂ ℝ: ξ < ω₁⟩$ of compact sets and $⟨W_ξ ∈ [ω]^ω: ξ < ω₁⟩$ such that $ℝ = ⋃_{ξ<ω₁}P_ξ$ and for every ξ < ω₁, $⟨fₙ ↾ P_ξ⟩_{n∈W_ξ}$ is a monotone uniformly convergent sequence of uniformly continuous functions. (e) Total failure of Martin's Axiom: 𝔠 > ω₁ and for every non-trivial ccc forcing ℙ there exist ω₁ dense sets in ℙ such that no filter intersects all of them
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We formulate a Covering Property Axiom $CPA_{cube}^{game}$, 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 < 𝔠, since every γ-set is universally null, while $CPA_{cube}^{game}$ implies that every universally null has cardinality less than 𝔠 = ω₂. We also show that $CPA_{cube}^{game}$ implies the existence of a partition of ℝ into ω₁ null compact sets.
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