### Parametrized Cichoń's diagram and small sets

Janusz Pawlikowski, Ireneusz Recław (1995)

Fundamenta Mathematicae

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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 ${G}_{\delta}$ 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 ${F}_{\sigma}$ and ${M}_{x}:x\in {w}^{w}$ is a basis for the ideal of meager subsets of ${2}^{w}$; •$\forall x,y{N}_{e\left(x\right)}\subseteq {N}_{y}\Rightarrow {M}_{x}\subseteq {M}_{f\left(y\right)}$. From this we derive that for a separable metric space X, •if for all Borel (resp. ${G}_{\delta}$) sets...