Covering Property Axiom $CPA_{cube}$ and its consequences

Krzysztof Ciesielski; Janusz Pawlikowski

Covering Property Axiom $C P A_{c u b e}$ and its consequences

Krzysztof Ciesielski; Janusz Pawlikowski

Fundamenta Mathematicae (2003)

Volume: 176, Issue: 1, page 63-75
ISSN: 0016-2736

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Abstract

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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 \in ℝ^{ℝ} ⟩_{n < ω}

of Borel functions there are sequences:

⟨ P_{ξ} \subset ℝ : ξ < ω ₁ ⟩

of compact sets and

⟨ W_{ξ} \in {[ω]}^{ω} : ξ < ω ₁ ⟩

such that

ℝ = ⋃_{ξ < ω ₁} P_{ξ}

and for every ξ < ω₁,

⟨ f ₙ ↾ P_{ξ} ⟩_{n \in W_{ξ}}

is a monotone uniformly convergent sequence of uniformly continuous functions. (e) Total failure of Martin’s Axiom: > 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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Krzysztof Ciesielski, and Janusz Pawlikowski. "Covering Property Axiom $CPA_{cube}$ and its consequences." Fundamenta Mathematicae 176.1 (2003): 63-75. <http://eudml.org/doc/283393>.

@article{KrzysztofCiesielski2003,
abstract = {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) cof() = ω₁ < , i.e., the cofinality of the measure ideal 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: > ω₁ and for every non-trivial ccc forcing ℙ there exist ω₁ dense sets in ℙ such that no filter intersects all of them},
author = {Krzysztof Ciesielski, Janusz Pawlikowski},
journal = {Fundamenta Mathematicae},
keywords = {continuous images; perfectly meager; universally null; cofinality of null ideal; uniform convergence; Martin's Axiom},
language = {eng},
number = {1},
pages = {63-75},
title = {Covering Property Axiom $CPA_\{cube\}$ and its consequences},
url = {http://eudml.org/doc/283393},
volume = {176},
year = {2003},
}

TY - JOUR
AU - Krzysztof Ciesielski
AU - Janusz Pawlikowski
TI - Covering Property Axiom $CPA_{cube}$ and its consequences
JO - Fundamenta Mathematicae
PY - 2003
VL - 176
IS - 1
SP - 63
EP - 75
AB - 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) cof() = ω₁ < , i.e., the cofinality of the measure ideal 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: > ω₁ and for every non-trivial ccc forcing ℙ there exist ω₁ dense sets in ℙ such that no filter intersects all of them
LA - eng
KW - continuous images; perfectly meager; universally null; cofinality of null ideal; uniform convergence; Martin's Axiom
UR - http://eudml.org/doc/283393
ER -

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