Selection Principles Milestone Problems — Problem 10 ([AddQuad])

Definitions and mathematical statement

The supplied question is \[ \mathsf{add}\left(\mathsf{U}_{\mathrm{fin}}(\Gamma,\mathcal O)\right)=\mathfrak b\;? \] The record does not specify the underlying class of spaces or its cover conventions. The following gives a conditional formulation using countable open covers; these conventions require confirmation against the source.

For a topological space $Y$, let $\mathcal O(Y)$ denote the countable families of open subsets of $Y$ that cover $Y$, with $Y$ itself excluded as a member. Let $\Gamma(Y)$ consist of the countably infinite members $\mathcal U$ of $\mathcal O(Y)$ such that each point of $Y$ belongs to all but finitely many members of $\mathcal U$. The property $\mathsf{U}_{\mathrm{fin}}(\Gamma,\mathcal O)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ from $\Gamma(Y)$, each having no finite subcover, there exist finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal F_n:n\in{\mathbb{N}}\right\}\in\mathcal O(Y). \] Write $\mathcal P$ for this property.

To specify the union operation in the additivity number, fix an ambient topological space $E$ and give every subset of $E$ its subspace topology. Define, whenever a witnessing family exists, \[ \mathsf{add}_E(\mathcal P) =\min\left\{|I|:\ X_i\subseteq E\text{ and }X_i\in\mathcal P\ (i\in I),\quad \bigcup_{i\in I}X_i\notin\mathcal P\right\}. \] Here $X\in\mathcal P$ means that $X$ has the stated selection property. For $f,g\in{\mathbb{N}}^{{\mathbb{N}}}$, write $f\leq^*g$ if $f(n)\leq g(n)$ for all but finitely many $n$. The bounding number is \[ \mathfrak b=\min\left\{|B|:B\subseteq{\mathbb{N}}^{{\mathbb{N}}},\quad \forall g\in{\mathbb{N}}^{{\mathbb{N}}}\ \exists f\in B\ (f\not\leq^*g)\right\}. \] The intended problem asks whether the additivity number of $\mathcal P$ equals $\mathfrak b$. Under the conditional ambient-space formulation above, its target is $\mathsf{add}_E(\mathcal P)=\mathfrak b$ for the intended ambient space and admissible subspaces. The supplied record does not identify these, so this is not an assertion or question quantified over all ambient spaces $E$. Any restrictions on admissible subspaces must also be incorporated into the definition of additivity.

Short English statement

Does the least number of spaces with the selection property $\mathsf{U}_{\mathrm{fin}}(\Gamma,\mathcal O)$ whose union fails that property equal the bounding number $\mathfrak b$? The intended class of spaces and cover conventions remain unspecified in the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $800638$, exact source problem code catalogue entry, title "Selection Principles Milestone Problems --- Problem 10 ([AddQuad])." The record attributes the problem to Tsaban, with proposed year 2003.
• Supplied source: Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 10. Supplied URL: https://arxiv.org/abs/math/0312182. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not an independently verified visit.
• The source excerpt places this question within the general problem of expressing additivity numbers of selection properties through cardinal characteristics such as $\mathfrak b$ and $\mathfrak d$, and describes it as one of three particularly interesting cases. It refers to [LecceSurvey] for a list of properties with open additivity questions. No bibliographic identification for [LecceSurvey] is supplied.
• The supplied attribution/context label is [AddQuad]; its bibliographic identification is not supplied. The registry labels this problem "open"; that label is supplied curation, not an independently verified assessment of its present status.