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 $B\subseteq X\times 2^w$ with all...

We prove the results stated in the title.

A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.

Under Martin’s axiom, collapsing of the continuum by Sacks forcing $𝕊$ is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if $𝔡=𝔠$ proving that under this hypothesis there are no small uncountable maximal antichains in $𝕊$. We also construct a partition of ${}^{\omega }2$ into $𝔠$ perfect sets which is a maximal antichain in $𝕊$ and show that $s^0$-sets are exactly (subsets of) selectors of maximal antichains of perfect sets.

We investigate properties of permitted trigonometric thin sets and construct uncountable permitted sets under some set-theoretical assumptions.

By an ${\omega }_1$- tree we mean a tree of power ${\omega }_1$ and height ${\omega }_1$. Under CH and $2^{{\omega }_1}>{\omega }_2$ we call an ${\omega }_1$-tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between ${\omega }_1$ and $2^{{\omega }_1}$. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus $2^{{\omega }_1}>{\omega }_2$ that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus $2^{{\omega }_1}={\omega }_4$ that there only exist Kurepa trees with ${\omega }_3$-many branches, which answers another...

Les travaux récents de Woodin ont considérablement renouvelé la théorie des ensembles en lui apportant une intelligibilité globale et en restaurant son unité. Pour la première fois, ses résultats ouvrent une perspective réaliste de résoudre le problème du continu, et, à tout le moins, ils établissent le caractère irréfutablement signifiant et précis de celui-ci.