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## Parametrized Cichoń's diagram and small sets

### Volume 147 / 1995

Fundamenta Mathematicae 147 (1995), 135-155 DOI: 10.4064/fm-147-2-135-155

#### Abstract

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

#### Authors

• Janusz Pawlikowski
• Ireneusz Recław

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