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

Krzysztof Ciesielski; Andrés Millán; Janusz Pawlikowski

Fundamenta Mathematicae (2003)

  • Volume: 176, Issue: 2, page 143-155
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Krzysztof Ciesielski, Andrés Millán, and Janusz Pawlikowski. "Uncountable γ-sets under axiom $CPA_{cube}^{game}$." Fundamenta Mathematicae 176.2 (2003): 143-155. <http://eudml.org/doc/283056>.

@article{KrzysztofCiesielski2003,
abstract = {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 < , since every γ-set is universally null, while $CPA_\{cube\}^\{game\}$ implies that every universally null has cardinality less than = ω₂. We also show that $CPA_\{cube\}^\{game\}$ implies the existence of a partition of ℝ into ω₁ null compact sets.},
author = {Krzysztof Ciesielski, Andrés Millán, Janusz Pawlikowski},
journal = {Fundamenta Mathematicae},
keywords = {-set; strongly meager sets; covering property axiom; iterated perfect set model},
language = {eng},
number = {2},
pages = {143-155},
title = {Uncountable γ-sets under axiom $CPA_\{cube\}^\{game\}$},
url = {http://eudml.org/doc/283056},
volume = {176},
year = {2003},
}

TY - JOUR
AU - Krzysztof Ciesielski
AU - Andrés Millán
AU - Janusz Pawlikowski
TI - Uncountable γ-sets under axiom $CPA_{cube}^{game}$
JO - Fundamenta Mathematicae
PY - 2003
VL - 176
IS - 2
SP - 143
EP - 155
AB - 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 < , since every γ-set is universally null, while $CPA_{cube}^{game}$ implies that every universally null has cardinality less than = ω₂. We also show that $CPA_{cube}^{game}$ implies the existence of a partition of ℝ into ω₁ null compact sets.
LA - eng
KW - -set; strongly meager sets; covering property axiom; iterated perfect set model
UR - http://eudml.org/doc/283056
ER -

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