Parametrized Cichoń's diagram and small sets

Pawlikowski, Janusz; Recław, Ireneusz

Fundamenta Mathematicae, Tome 146 (1995), p. 135-155 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $w^{w} \times 2^{w}$ and continuous functions $e, f : w^{w} \to w^{w}$ such that • N is $G_{δ}$ and $N_{x} : x \in w^{w}$ , 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 $M_{x} : x \in w^{w}$ is a basis for the ideal of meager subsets of $2^{w}$ ; • $\forall x, y N_{e (x)} \subseteq N_{y} \Rightarrow M_{x} \subseteq M_{f (y)}$ . From this we derive that for a separable metric space X, •if for all Borel (resp. $G_{δ}$ ) sets $B \subseteq X \times 2^{w}$ with all vertical sections null, $\cup_{x \in X} B_{x}$ is null, then for all Borel (resp. $F_{σ}$ ) sets $B \subseteq X \times 2^{w}$ with all vertical sections meager, $\cup_{x \in X} B_{x}$ is meager; •if there exists a Borel (resp. a “nice” $G_{δ}$ ) set $B \subseteq X \times 2^{w}$ such that $B_{x} : x \in X$ is a basis for measure zero sets, then there exists a Borel (resp. $F_{σ}$ ) set $B \subseteq X \times 2^{w}$ such that $B_{x} : x \in X$ is a basis for meager sets

Publié le : 1995-01-01

Zbl 0847.04004

EUDML-ID : urn:eudml:doc:212079

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     author = {Janusz Pawlikowski and Ireneusz Rec\l aw},
     title = {Parametrized Cicho\'n's diagram and small sets},
     journal = {Fundamenta Mathematicae},
     volume = {146},
     year = {1995},
     pages = {135-155},
     zbl = {0847.04004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv147i2p135bwm}
}

Pawlikowski, Janusz; Recław, Ireneusz. Parametrized Cichoń's diagram and small sets. Fundamenta Mathematicae, Tome 146 (1995) pp. 135-155. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv147i2p135bwm/

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