Selection Principles in Mathematics — Problem 51 ([SFT])

Definitions and mathematical statement

For a topological space $T$, let $\mathcal O(T)$ be the family of open covers of $T$. An open $\omega$-cover of $T$ is a family $\mathcal U$ of open subsets of $T$ such that $T\notin\mathcal U$ and every finite subset of $T$ is contained in some member of $\mathcal U$. Write $\Omega(T)$ for the family of these covers. Under the standard selection-principle convention, $T$ satisfies $\mathsf U_{\mathrm{fin}}(\mathcal O,\Omega)$ if, for every sequence $(\mathcal U_n)_{n\in\mathbb N}$ of open covers of $T$, each having no finite subcover, there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal F_n:n\in\mathbb N\right\}\in\Omega(T). \] For functions $f,g\in\mathbb N^{\mathbb N}$, define eventual domination by \[ g\leq^* f\quad\Longleftrightarrow\quad \{n\in\mathbb N:g(n)>f(n)\}\text{ is finite}. \] For $k\geq1$, the pointwise maximum of $f_1,\ldots,f_k\in\mathbb N^{\mathbb N}$ is the function $n\mapsto\max_{1\leq i\leq k}f_i(n)$.

The supplied Problem 51 assumes that $X$ satisfies $\mathsf U_{\mathrm{fin}}(\mathcal O,\Omega)$ and asks whether the set displayed in the source as \[ G=\{g\in\mathbb N:(\exists k)(\exists f_1,\ldots,f_k\in Y)\ g\leq^*\max\{f_1,\ldots,f_k\}\} \] also satisfies $\mathsf U_{\mathrm{fin}}(\mathcal O,\Omega)$.

This formulation is incomplete: $Y$ is undefined, its relationship to $X$ is unspecified, and no topology on either space is supplied. Moreover, the membership condition $g\in\mathbb N$ does not provide a function to which the usual eventual-domination relation applies. If the intended membership condition is $g\in\mathbb N^{\mathbb N}$, if $Y\subseteq\mathbb N^{\mathbb N}$, and if only nonempty finite maxima are intended, then the corresponding well-defined set is \[ G_Y=\left\{g\in\mathbb N^{\mathbb N}:\exists k\in\mathbb N,\ k\geq1,\ \exists f_1,\ldots,f_k\in Y,\ g\leq^*\max_{1\leq i\leq k}f_i\right\}. \] These are conditional interpretations, not supplied hypotheses. The record does not determine which relationship between $X$ and $Y$, or which topology on $G_Y$, is intended, so it does not yet specify a precise implication to be assessed.

Short English statement

The source asks whether a set obtained by closing $Y$ downward under eventual domination by finite pointwise maxima inherits $\mathsf U_{\mathrm{fin}}(\mathcal O,\Omega)$ from $X$. The relationship between $X$ and $Y$, the relevant topologies, and the apparent function-space membership error remain unresolved. The supplied registry labels this problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $800622$, source code catalogue entry, title "Selection Principles in Mathematics --- Problem 51 ([SFT])." The record attributes the problem to Tsaban, with proposed year 2006, and supplies the status label "open."
• Supplied source attribution: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 51. Supplied URL: https://arxiv.org/abs/math/0609601. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not independently verified access or source verification.
• The supplied attribution/context marker is [SFT]; its bibliographic expansion is absent from the record.