Selection Principles in Mathematics — Problem 44 ([BG])

Definitions and mathematical statement

Let ${\mathbb{N}}=\{0,1,2,\ldots\}$, and give ${\mathbb{Z}}^{{\mathbb{N}}}$ coordinatewise addition and the product topology of the discrete topology on ${\mathbb{Z}}$. For an explicit conventional metric, put \[ d(x,y)=\begin{cases}0,&x=y,\\2^{-\min\{n\in{\mathbb{N}}:x(n)\ne y(n)\}},&x\ne y.\end{cases} \] A subgroup $G\le{\mathbb{Z}}^{{\mathbb{N}}}$ carries the induced topology and metric. It has strong measure zero if, for every sequence $(\varepsilon_n)_{n\in{\mathbb{N}}}$ of positive real numbers, there are sets $A_n\subseteq G$ such that \[ G=\bigcup_{n\in{\mathbb{N}}}A_n\quad\text{and}\quad \operatorname{diam}_d(A_n)<\varepsilon_n\quad(n\in{\mathbb{N}}), \] where the diameter of the empty set is taken to be zero.

Write $\mathcal O(G)$ for the family of open covers of $G$. The property $\mathsf S_{\mathrm{fin}}(\mathcal O,\mathcal O)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ in $\mathcal O(G)$, there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ with \[ G=\bigcup_{n\in{\mathbb{N}}}\bigcup\mathcal F_n. \] The property $\mathsf S_1(\mathcal O,\mathcal O)$ means that for every such sequence there are choices $U_n\in\mathcal U_n$ satisfying $G=\bigcup_{n\in{\mathbb{N}}}U_n$.

For an explicit eventual-domination interpretation, call $G$ unbounded if \[ \forall h\in{\mathbb{N}}^{{\mathbb{N}}}\ \exists g\in G\quad \left|\{n\in{\mathbb{N}}:|g(n)|>h(n)\}\right|=\infty. \] For subgroups this is equivalent to having no eventual upper bound in ${\mathbb{Z}}^{{\mathbb{N}}}$ under the coordinatewise eventual order. This notion is not metric unboundedness.

With these conventions, the question is whether it is consistent with $\mathsf{ZFC}$ that there exists a subgroup $G\le{\mathbb{Z}}^{{\mathbb{N}}}$ which has strong measure zero, satisfies $\mathsf S_{\mathrm{fin}}(\mathcal O,\mathcal O)$, is unbounded in the sense above, and fails $\mathsf S_1(\mathcal O,\mathcal O)$. Explicitly, the last failure requires a sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ of open covers of $G$ for which every choice $U_n\in\mathcal U_n$ satisfies $\bigcup_n U_n\ne G$.

The supplied excerpt does not specify the metric, the meaning of unboundedness, or the background theory for consistency. The formulation above makes conventional choices explicit; their agreement with the intended source formulation requires review.

Short English statement

Can a model of set theory contain a subgroup of ${\mathbb{Z}}^{{\mathbb{N}}}$ that has strong measure zero and the finite open-cover selection property, but is unbounded under eventual domination and fails the one-element open-cover selection property? The metric, unboundedness, and background-theory conventions used here require confirmation from the source context.

Sources

• Local UnsolvedMath registry, permanent ID $800616$, exact source problem code catalogue entry; supplied title: Selection Principles in Mathematics --- Problem 44 ([BG]). Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 44. Supplied source URL: https://arxiv.org/abs/math/0609601. The record identifies the extraction as arxiv_latex and supplies the access date 2026-10-16; these are supplied provenance details, not independently verified access.
• The registry lists Tsaban as proposer and 2006 as the proposed year, while the problem title and background retain the attribution/context marker [BG]. No expansion or bibliography entry for that marker is supplied.
• The registry labels the problem open. This is a supplied status label, not an independently verified literature assessment.