Remarks on small sets of reals

We show that the following are consistent with ZFC: 1. Strongly meager sets form an ideal with the same additivity as the ideal of meager sets. 2. There exists a strong measure zero set of size > d (dominating number).

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.

We will show that there is no ZFC example of a set distinguishing between universally null and perfectly meager sets.

We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A intersect C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.

We show that some set-theoretic assumptions (for example Martin's Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets in (in R^n). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the sigma-algebra of sets with the property of Baire.

We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.

We show that if the real line is the disjoint union of kappa meager sets such that every meager set is contained in a countable union of them, then kappa = omega_1. This answers a question addressed by J. Cichon. We also prove two theorems saying roughly that any attempt to produce the isomorphism type of the meager ideal in the Cohen real and the random real extensions must fail. All our results hold for "meager" replaced by "null", as well.

Our aim is to investigate spaces with sigma-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of Tkachuk [30], Hutchinson [17] and the authors of [8].

We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped continuously onto the Cantor space. These results allow to separate the properties of Menger and $\mathsf{S}_1(\Gamma,\mathrm{O})$ in the realm of sets of reals without perfect subsets and solve a problem of Nowik and Tsaban concerning perfec...

We ask whether $\mathbf{\Delta^1_2}$ or $\mathbf{\Sigma^1_2}$ equivalence relations with $I$-small classes for $I$ a $\sigma$-ideal must have perfectly many classes. We show that for a wide class of ccc $\sigma$-ideals, a positive answer for $\mathbf{\Delta^1_2}$ equivalence relations is equivalent to the $I$-measurability of $\mathbf{\Delta^1_2}$ sets. However, the analogous statement for $\mathbf{\Sigma^1_2}$ equivalence relations is false: $\mathbf{\Sigma^1_2}$ equivalence...