Selection Principles Milestone Problems — Problem 3 ([ideals, Problem 17], [o-bdd, Problem 2])

Definitions and mathematical statement

The supplied statement has two unresolved prerequisites: its displayed ambient set $\mathbb N$ cannot literally contain the strictly increasing functions under discussion, and neither $Q$ nor the topology on $H=B\cup Q$ is defined. The following gives a conditional formulation using the interpretation that the elements of $B$ belong to $\mathbb N^{\mathbb N}$. This interpretation requires confirmation, and no definition of $Q$ is inferred.

For $f,g\in\mathbb N^{\mathbb N}$, define eventual domination by \[ f\leq^*g\quad\Longleftrightarrow\quad \{n\in\mathbb N:f(n)>g(n)\}\text{ is finite}. \] A family $A\subseteq\mathbb N^{\mathbb N}$ is unbounded if \[ \forall g\in\mathbb N^{\mathbb N}\ \exists f\in A\quad f\not\leq^*g. \] The bounding number is \[ \mathfrak b=\min\{|A|:A\subseteq\mathbb N^{\mathbb N}\text{ is unbounded}\}. \] A $\mathfrak b$-scale as described in the supplied record is an unbounded family \[ B=\{f_\alpha:\alpha<\mathfrak b\}\subseteq\mathbb N^{\mathbb N} \] of strictly increasing functions, meaning $f_\alpha(n)<f_\alpha(n+1)$ for every $n\in\mathbb N$, such that \[ \alpha<\beta<\mathfrak b\quad\Longrightarrow\quad f_\alpha\leq^*f_\beta. \]

For a topological space $X$, let $\mathcal O(X)$ be the family of open covers of $X$, with the convention that a cover does not contain $X$ itself. An open $\gamma$-cover is an infinite family $\mathcal U$ of open subsets of $X$ such that \[ \forall x\in X\quad\{U\in\mathcal U:x\notin U\}\text{ is finite}. \] Write $\Gamma(X)$ for the family of these covers. The property $\mathsf S_1(\Gamma,\Gamma)$ means \[ \forall (\mathcal U_n)_{n\in\mathbb N}\in\Gamma(X)^{\mathbb N}\quad \exists (U_n)_{n\in\mathbb N}\quad \left[(\forall n\in\mathbb N\; U_n\in\mathcal U_n) \ \land\ \{U_n:n\in\mathbb N\}\in\Gamma(X)\right]. \] In particular, the selected family must be infinite as a set of distinct open subsets.

For the contextual assertion, $\mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma)$ means that for every sequence $(\mathcal U_n)_{n\in\mathbb N}$ of open covers, each having no finite subcover, there are finite families $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal F_n:n\in\mathbb N\right\}\in\Gamma(X). \]

Question, conditional on completing the source definitions. For the $\mathfrak b$-scale $B$ described above, form $H=B\cup Q$, with $Q$ and the topology on $H$ to be recovered from the cited source. Must $H$ satisfy $\mathsf S_1(\Gamma,\Gamma)$? Equivalently, must every sequence of open $\gamma$-covers of that space admit one selection from each cover whose selected family is again an open $\gamma$-cover?

The supplied record attributes to reference [ideals] the assertion that $H^k$ satisfies $\mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma)$ for every positive integer $k$, where finite powers carry the product topology. This is retained as supplied background, subject to the missing definitions and hypotheses; it is not independently verified here. The record also leaves unclear whether the question concerns every such scale or a particular scale fixed in preceding source context.

Short English statement

The question asks whether the space $H=B\cup Q$, constructed from a $\mathfrak b$-scale, has the property that one can choose a single member of each open $\gamma$-cover in any sequence so that the chosen family is again a $\gamma$-cover. A precise instance requires the missing definition of $Q$, the topology, and confirmation of the intended function space and scale quantifier. The supplied background attributes a weaker finite-selection property to all finite powers of $H$.

Sources

• Local UnsolvedMath registry, permanent ID $800631$, exact source problem code catalogue entry; supplied title: "Selection Principles Milestone Problems --- Problem 3 ([ideals, Problem 17], [o-bdd, Problem 2])." The registry attributes the problem to Tsaban, with proposed year 2003, and labels its status as open. These are supplied registry labels, not independently verified conclusions.
• Supplied source list: Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 3. Supplied URL: https://arxiv.org/abs/math/0312182. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance details are supplied and do not represent source access performed for this draft.
• Supplied attribution/context: [ideals, Problem 17] and [o-bdd, Problem 2]. The supplied statement attributes the finite-power $\mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma)$ result to [ideals]. Full bibliographic entries for these reference labels are absent from the supplied record.