Selection Principles in Mathematics — Problem 16 (Krawczyk)

Definitions and mathematical statement

Give each set $X\subseteq{\mathbb{R}}$ its usual subspace topology. An open $\omega$-cover of $X$ is a family $\mathcal U$ of open subsets of $X$ such that $X\notin\mathcal U$ and every finite subset of $X$ is contained in some member of $\mathcal U$. Write $\Omega(X)$ for the collection of these covers. An open $\gamma$-cover is an infinite family of open subsets of $X$ such that each point of $X$ belongs to all but finitely many members; write $\Gamma(X)$ for the collection of these covers. The property $\mathsf S_1(\Omega,\Gamma)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ from $\Omega(X)$, there are choices $U_n\in\mathcal U_n$ such that $\{U_n:n\in{\mathbb{N}}\}\in\Gamma(X)$. In particular, the selected family must be infinite.

For a metric space $(Z,d)$ and $s>0$, define \[ \mathcal H_d^s(Z)=\lim_{\delta\downarrow0}\inf\left\{\sum_{i=1}^{\infty}(\operatorname{diam}_d E_i)^s: Z\subseteq\bigcup_{i=1}^{\infty}E_i,\ E_i\subseteq Z,\ \operatorname{diam}_d E_i\leq\delta\right\}, \qquad \dim_H Z=\inf\{s>0:\mathcal H_d^s(Z)=0\}, \] with $\inf\varnothing=+\infty$. This convention gives the empty space dimension zero. For $X\subseteq{\mathbb{R}}$ and a metric space $(Y,d_Y)$, equip the product with \[ d_{X\times Y}((x,y),(x',y'))=\sqrt{|x-x'|^2+d_Y(y,y')^2}. \] When $Y\subseteq{\mathbb{R}}$ has its usual metric, this is the Euclidean metric on $X\times Y\subseteq{\mathbb{R}}^2$.

The supplied question asks whether it is consistent, relative to $\mathsf{ZFC}$, that uncountable sets of reals with $\mathsf S_1(\Omega,\Gamma)$ exist and that every such set has Hausdorff-dimension-zero product with every admissible Hausdorff-dimension-zero set $Y$. The source does not specify the domain or metric of $Y$. To retain this unresolved qualification, let $\mathcal D$ denote a specified class of admissible metric sets. For that specification, let $P_{\mathcal D}$ be the conjunction \[ \begin{gathered} \exists X\subseteq{\mathbb{R}}\;\bigl(|X|>\aleph_0\ \land\ X\models\mathsf S_1(\Omega,\Gamma)\bigr),\\ \forall X\subseteq{\mathbb{R}}\;\left[ \bigl(|X|>\aleph_0\ \land\ X\models\mathsf S_1(\Omega,\Gamma)\bigr) \Longrightarrow \forall Y\in\mathcal D\;\bigl(\dim_H Y=0\Longrightarrow\dim_H(X\times Y)=0\bigr) \right]. \end{gathered} \] The relative-consistency question is whether \[ \operatorname{Con}(\mathsf{ZFC})\Longrightarrow \operatorname{Con}(\mathsf{ZFC}+P_{\mathcal D}). \] This is a formulation schema until the intended class $\mathcal D$ is fixed. Taking $\mathcal D$ to be all subsets of ${\mathbb{R}}$ with their usual metrics gives a precise real-set interpretation, but the supplied record does not establish that this is the intended restriction. All quantifiers in $P_{\mathcal D}$ are internal to the model in the consistency question.

Short English statement

The question asks whether, consistently with $\mathsf{ZFC}$, uncountable sets of reals satisfying $\mathsf S_1(\Omega,\Gamma)$ exist while every such set has a Hausdorff-dimension-zero product with every admissible set of Hausdorff dimension zero. The intended domain and metric for the second factor $Y$ remain unspecified. The registry labels the problem open; this is supplied curation, not an independently verified status assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800590$, exact source code catalogue entry, in the AMR Open Problem Lists collection. The record labels the problem open, lists Tsaban as proposer and $2006$ as the proposed year, and gives Krawczyk as the attribution/context.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 16. Supplied source link: https://arxiv.org/abs/math/0609601. The background identifies this item with Krawczyk.
• The supplied provenance reports access on 2026-10-16 and extraction by arxiv_latex. These are source-record provenance claims; the source and its mathematical status have not been independently checked for this draft.