Selection Principles in Mathematics — Problem 12 ([CBC])

Definitions and mathematical statement

Let $X$ be a topological space. An $\omega$-cover of $X$ is a family $\mathcal U$ of subsets of $X$ such that $X\notin\mathcal U$ and every finite subset of $X$ is contained in some member of $\mathcal U$. Under the standard countable-cover convention, write $\mathrm B_\Omega(X)$ for the family of countable $\omega$-covers whose members are Borel subsets of $X$. Here the Borel subsets form the sigma-algebra generated by the open subsets of $X$.

For families $\mathcal A$ and $\mathcal B$ of covers of a space, the selection principle $\mathsf S_1(\mathcal A,\mathcal B)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ of members of $\mathcal A$, there are choices $U_n\in\mathcal U_n$ such that $\{U_n:n\in{\mathbb{N}}\}\in\mathcal B$. The principle $\mathsf S_{\mathrm{fin}}(\mathcal A,\mathcal B)$ means that for every such sequence there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that $\bigcup_{n\in{\mathbb{N}}}\mathcal F_n\in\mathcal B$. Set \[ P_1(X):=\mathsf S_1(\mathrm B_\Omega(X),\mathrm B_\Omega(X)),\qquad P_{\mathrm{fin}}(X):=\mathsf S_{\mathrm{fin}}(\mathrm B_\Omega(X),\mathrm B_\Omega(X)). \] For a positive integer $m$, give $X^m$ the product topology, and interpret its Borel covers relative to that topology.

Question, relative to the intended class of spaces. For each $i\in\{1,\mathrm{fin}\}$, does the implication \[ \forall X\in\mathcal C\;\forall m\in{\mathbb{Z}}_{\geq1},\qquad P_i(X)\Longrightarrow P_i(X^m) \] hold, where $\mathcal C$ denotes the intended ambient class of spaces? These are two separate preservation questions; an affirmative answer to either would answer the source's question whether any of the two classes is preserved by finite powers.

Unresolved scope. The supplied excerpt does not specify $\mathcal C$ or explicitly define $\mathrm B_\Omega$. The source title provides a context of special sets of reals, but does not by itself determine the exact hypotheses. The definitions above use the standard interpretation as countable Borel $\omega$-covers and make the preservation question precise relative to $\mathcal C$; the intended domain and cover convention require confirmation from the source.

Supplied background qualification. Write $\Omega$ for the corresponding family of countable open $\omega$-covers, and $\Gamma$ for the countably infinite open covers in which each point belongs to all but finitely many members. The supplied record states, with reference label [coc2], that $\mathsf S_1(\Omega,\Gamma)$, $\mathsf S_1(\Omega,\Omega)$, and $\mathsf S_{\mathrm{fin}}(\Omega,\Omega)$ are the only properties in the open case preserved under finite powers. This is retained as a source-reported contextual assertion: the excerpt does not specify the collection of properties over which its word "only" ranges.

Short English statement

For each of two Borel cover selection properties, ask whether every positive finite Cartesian power of a space with that property also has it. The intended class of spaces and the countable Borel cover convention need source confirmation. The registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry: permanent ID $800586$, exact source problem code catalogue entry, title "Selection Principles in Mathematics --- Problem 12 ([CBC])." The supplied record lists Tsaban as proposer, gives the year 2006, and labels the problem open. Its upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied source reference: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 12, https://arxiv.org/abs/math/0609601. The record reports extraction by arxiv_latex and an access date of 2026-10-16. These are supplied provenance details; this draft does not claim to have accessed or independently verified the source.
• The supplied attribution/context marker is [CBC]. The contextual assertion about open covers carries reference label [coc2]. Neither label is expanded bibliographically in the supplied record, so no author identities or additional references are inferred.