Selection Principles in Mathematics — Problem 3 ([ideals])

Definitions and mathematical statement

Give each set $X\subseteq{\mathbb{R}}$ its subspace topology. We use the following standard cover conventions, which are not explicitly specified in the supplied excerpt. A countable open cover of $X$ is a countable family $\mathcal U$ of relatively open subsets of $X$ such that $\bigcup\mathcal U=X$ and $X\notin\mathcal U$. An $\omega$-cover is such a cover for which every finite subset of $X$ is contained in some member of $\mathcal U$. A $\gamma$-cover is an infinite such cover for which each point of $X$ belongs to all but finitely many members. Write $\Omega(X)$ and $\Gamma(X)$ for these classes, respectively.

The property $\mathsf S_1(\Gamma,\Gamma)$ means that \[ \forall (\mathcal U_n)_{n\in{\mathbb{N}}}\in\Gamma(X)^{{\mathbb{N}}}\quad \exists (U_n)_{n\in{\mathbb{N}}}\quad \left[(\forall n\in{\mathbb{N}})\ U_n\in\mathcal U_n \ \text{ and }\ \{U_n:n\in{\mathbb{N}}\}\in\Gamma(X)\right]. \] For the background comparison, $\mathsf S_{\mathrm{fin}}(\Omega,\Omega)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ of members of $\Omega(X)$, there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that $\bigcup_{n\in{\mathbb{N}}}\mathcal F_n\in\Omega(X)$.

For $f,g\in{\mathbb{N}}^{{\mathbb{N}}}$, write $f\leq^*g$ when $f(n)\leq g(n)$ for all but finitely many $n$. Define the bounding and dominating numbers by \[ \mathfrak b=\min\left\{|B|:B\subseteq{\mathbb{N}}^{{\mathbb{N}}},\ (\forall g\in{\mathbb{N}}^{{\mathbb{N}}})(\exists f\in B)\ f\not\leq^*g\right\}, \] \[ \mathfrak d=\min\left\{|D|:D\subseteq{\mathbb{N}}^{{\mathbb{N}}},\ (\forall g\in{\mathbb{N}}^{{\mathbb{N}}})(\exists f\in D)\ g\leq^*f\right\}. \] For infinite subsets $A,B\subseteq{\mathbb{N}}$, write $A\subseteq^*B$ if $A\setminus B$ is finite. The tower number $\mathfrak t$ is the least cardinal $\kappa$ for which there exists a sequence $(T_\alpha)_{\alpha<\kappa}$ of infinite subsets of ${\mathbb{N}}$, strictly decreasing modulo finite sets, with no infinite pseudointersection. Thus, for $\alpha<\beta<\kappa$, one has $T_\beta\subseteq^*T_\alpha$ and $T_\alpha\not\subseteq^*T_\beta$, and there is no infinite $A\subseteq{\mathbb{N}}$ with $A\subseteq^*T_\alpha$ for every $\alpha<\kappa$. The notation $\operatorname{cf}(\kappa)$ denotes the least cardinality of an unbounded subset of the ordinal $\kappa$. For a property $P$ of sets of reals, put \[ \operatorname{non}(P)=\min\{|Y|:Y\subseteq{\mathbb{R}}\text{ and }Y\text{ does not satisfy }P\}. \]

Problem. Does ZFC prove the existence of a set $X\subseteq{\mathbb{R}}$ such that \[ |X|=\mathfrak b \qquad\text{and}\qquad X\models\mathsf S_1(\Gamma,\Gamma)? \] Here $\mathfrak b$ is computed in the same set-theoretic universe as $X$.

Supplied historical context. The source states that dichotomic arguments give, in ZFC, a set of reals satisfying $\mathsf S_1(\Gamma,\Gamma)$ of cardinality $\mathfrak t$, citing [alpha_i], and a set of reals satisfying $\mathsf S_{\mathrm{fin}}(\Omega,\Omega)$ of cardinality $\operatorname{cf}(\mathfrak d)$, citing [SFH]. It also states \[ \operatorname{non}(\mathsf S_1(\Gamma,\Gamma))=\mathfrak b, \qquad \operatorname{non}(\mathsf S_{\mathrm{fin}}(\Omega,\Omega))=\mathfrak d, \] and that it is consistent that $\mathfrak b>\mathfrak t$ and $\mathfrak d>\operatorname{cf}(\mathfrak d)$. These comparisons motivate the source's description of the preceding existence results as unsatisfactory. They are retained as supplied historical assertions, not as independently verified results or a current literature assessment. The exact cover conventions and the bibliography entries behind [alpha_i], [SFH], and the context label [ideals] require source review.

Short English statement

Does ZFC prove that some set of reals of cardinality exactly $\mathfrak b$ satisfies $\mathsf S_1(\Gamma,\Gamma)$? This means that from each cover in any sequence of open $\gamma$-covers one can select one member so that the selected family is again a $\gamma$-cover. The supplied registry labels the problem open; that label has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800577$, exact source problem code catalogue entry; collection: AMR Open Problem Lists. The record attributes the problem to Tsaban, with proposed year 2006, and labels its status open. These are supplied registry assertions.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 3. Supplied source URL: https://arxiv.org/abs/math/0609601. The record reports extraction by arxiv_latex and access on 2026-10-16; this is supplied provenance, not an access or verification performed for this draft.
• The supplied statement cites [alpha_i] for the $\mathfrak t$-sized example and [SFH] for the $\operatorname{cf}(\mathfrak d)$-sized example. No corresponding bibliography entries are included in the supplied record, so these labels cannot be resolved here.
• The supplied title is Selection Principles in Mathematics --- Problem 3 ([ideals]); the background also gives [ideals] as its attribution/context label. No bibliographic expansion of this label is supplied. The recorded upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.