Products of special sets of real numbers

Special sets of reals and weak forms of normality on Isbell-Mrówka spaces

We recall some classical results relating normality and some natural weakenings of normality in $Ψ$-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $λ$-sets and $σ$-sets. We introduce a new class of special sets of reals which corresponds the corresponding almost disjoint family of branches being $\aleph_0$-separated. This new class fits between $λ$-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $\mathcal A$ being potentially almost-normal (pseudonormal), in the sense that $\mathcal A$ is almo…

Selection Principles and special sets of reals: Open problems

We give a selection of major open problems involving selective properties, diagonalizations, and covering properties for sets of real numbers. This is a revision of the version published as a chapter in the book \textbf{Open Problems in Topology II} (E. Pearl, ed.), Elsevier B.V., 2007, 91--108. The present version reports solutions of some problems, uses up-to-date notation, and update bibliography. Comments and further updates would be appreciated.

关于$Δ$-集的乘积

On Products of $Δ$-sets

该数学研究证明了两项集合论一致性结论：存在平方非$Δ$-集的$Q$-集，以及若存在$Δ$-集则存在有限幂均为$Δ$-集的$Δ$-集。

Special subsets of the generalized Cantor space $2^κ$ and generalized Baire space $κ^κ$

In this paper, we are interested in parallels to the classical notions of special subsets in $\R$ defined in the generalized Cantor and Baire spaces ($2^κ$ and $κ^κ$). We consider generalizations of the well-known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, $σ$-sets, $γ$-sets, sets with Menger, Rothberger or Hurewicz property, but also of some less-know classes like $X$-small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller and Laver-null sets. We notice that many classical theorems regarding these classes c…

Finite powers and products of Menger sets

We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in the Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.

Products of Luzin-type sets with combinatorial properties

We construct Luzin-type subsets of the real line in all finite powers Rothberger, with a non-Menger product. To this end, we use a purely combinatorial approach which allows to weaken assumptions used earlier to construct sets with analogous properties. Our assumptions hold, e.g., in the Random model, where already known category theoretic methods fail.