# Parametrized Cichoń's diagram and small sets

Janusz Pawlikowski; Ireneusz Recław

Fundamenta Mathematicae (1995)

- Volume: 147, Issue: 2, page 135-155
- ISSN: 0016-2736

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topPawlikowski, Janusz, and Recław, Ireneusz. "Parametrized Cichoń's diagram and small sets." Fundamenta Mathematicae 147.2 (1995): 135-155. <http://eudml.org/doc/212079>.

@article{Pawlikowski1995,

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},

author = {Pawlikowski, Janusz, Recław, Ireneusz},

journal = {Fundamenta Mathematicae},

keywords = {measure zero set; Cichoń’s diagram; classes of small sets of reals; meager set},

language = {eng},

number = {2},

pages = {135-155},

title = {Parametrized Cichoń's diagram and small sets},

url = {http://eudml.org/doc/212079},

volume = {147},

year = {1995},

}

TY - JOUR

AU - Pawlikowski, Janusz

AU - Recław, Ireneusz

TI - Parametrized Cichoń's diagram and small sets

JO - Fundamenta Mathematicae

PY - 1995

VL - 147

IS - 2

SP - 135

EP - 155

AB - 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

LA - eng

KW - measure zero set; Cichoń’s diagram; classes of small sets of reals; meager set

UR - http://eudml.org/doc/212079

ER -

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