Selection Principles in Mathematics — Problem 15 ([Hdim])

Definitions and mathematical statement

Let ${\mathbb{N}}=\{1,2,\ldots\}$. The pseudointersection number $\mathfrak p$ is \[ \mathfrak p=\min\left\{|\mathcal A|:\mathcal A\subseteq[{\mathbb{N}}]^{\aleph_0},\ \text{every finite subfamily of }\mathcal A\text{ has infinite intersection, and no infinite }B\subseteq{\mathbb{N}}\text{ satisfies }B\setminus A\text{ finite for every }A\in\mathcal A\right\}. \] This is the conventional meaning of the otherwise undefined symbol $\mathfrak p$ in the supplied record.

For a metric space $(Z,d)$, Hausdorff dimension zero means that for every $s>0$, $\delta>0$, and $\varepsilon>0$, there is a countable cover $(A_k)_{k\geq1}$ of $Z$ such that \[ \operatorname{diam}_d(A_k)\leq\delta\quad(k\geq1),\qquad \sum_{k\geq1}\left(\operatorname{diam}_d(A_k)\right)^s<\varepsilon. \] Write this condition as $\dim_{\mathrm H}(Z,d)=0$.

Question (source Problem 15). Assume $|X|<\mathfrak p$. Is it true that for every admissible space $Y$, \[ \dim_{\mathrm H}Y=0\quad\Longrightarrow\quad\dim_{\mathrm H}(X\times Y)=0? \] The supplied record does not specify the ambient class of $X$ and $Y$, their metrics, or the product metric. Consequently, "admissible" remains unresolved: the displayed question is a statement template, not an assertion that the quantifier ranges over all metric spaces. Hausdorff dimension requires these metric choices, so a fully specified formulation requires recovery of the intended setting.

Supplied context. Conventionally, an open $\omega$-cover of $X$ is an open cover that does not contain $X$ itself and contains a member covering each finite subset of $X$. An open $\gamma$-cover is an infinite family of open sets such that each point of $X$ belongs to all but finitely many members. Let $\Omega$ and $\Gamma$ denote these respective classes. The property $\mathsf S_1(\Omega,\Gamma)$ means that from each sequence $(\mathcal U_n)_{n\geq1}$ of members of $\Omega$, one can select $U_n\in\mathcal U_n$ so that $\{U_n:n\geq1\}\in\Gamma$.

The record defines an open $n$-cover $\mathcal U$ by requiring that every $F\subseteq X$ with $|F|\leq n$ be contained in some member of $\mathcal U$. It describes $\mathsf S_1(\{\mathrm O_n\}_{n\in{\mathbb{N}}},\Gamma)$, called the strong $\gamma$-property and associated with reference labels [GM, strongdiags], as selection of $U_n\in\mathcal U_n$ from each sequence of open $n$-covers to obtain a $\gamma$-cover. The supplied text actually writes $\{\mathcal U_n:n\in{\mathbb{N}}\}$ as that resulting cover; the interpretation as $\{U_n:n\in{\mathbb{N}}\}$ is a proposed correction requiring review. Whether the open $n$-covers must exclude $X$ itself is also unstated.

For $Y$ of Hausdorff dimension zero, the source asserts that $\mathsf S_1(\Omega,\Gamma)$ for $X$ does not imply Hausdorff dimension zero of $X\times Y$, whereas the strong $\gamma$-property does imply it, citing [Hdim] for the latter assertion. These are supplied contextual claims in an unspecified metric setting, not independently verified results of this draft.

Short English statement

The question asks whether a set $X$ of cardinality below the pseudointersection number $\mathfrak p$ has a product of Hausdorff dimension zero with every eligible space $Y$ of Hausdorff dimension zero. The intended spaces and metrics are missing. The registry labels the problem open; that label and the contextual claims are supplied information requiring mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $800589$, exact source problem code catalogue entry, upstream ID $800589$, revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied record attributes the problem to Tsaban, with proposed year 2006, and labels its status open.
• Supplied source list: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 15. Supplied URL: https://arxiv.org/abs/math/0609601. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance details are supplied and were not independently verified.
• Supplied attribution/context: [Hdim]. The record cites [Hdim] for the product assertion involving the strong $\gamma$-property and [GM, strongdiags] for that property's terminology or definition. No bibliographic expansions or separate links for these labels are supplied.