Selection Principles in Mathematics — Problem 28 ([coc8])

Definitions and mathematical statement

The supplied record asks whether \[ \mathsf{S}_1(\Omega,\Omega)=\mathsf{S}_1(\Omega,\gimel(\Omega)). \] Here equality means equivalence of properties of spaces in the intended ambient class, rather than equality of families of covers. The record does not specify that ambient class, its cover conventions, or the definition of $\gimel$. Consequently, the precise scope of this question remains unresolved.

The following gives a conditional classical formulation using the identity supplied in the record. For this formulation, let $X$ be a topological space and use the convention that $\mathcal O(X)$ consists of countable families $\mathcal U$ of open subsets of $X$ such that $\bigcup\mathcal U=X$ and $X\notin\mathcal U$. Define \[ \Omega(X)=\left\{\mathcal U\in\mathcal O(X): (\forall F\subseteq X\text{ finite})(\exists U\in\mathcal U)\ F\subseteq U\right\}. \] These cover conventions are explicit provisional conventions; their agreement with the original source requires review.

For collections $\mathcal A(X)$ and $\mathcal B(X)$ of covers, the property $\mathsf S_1(\mathcal A,\mathcal B)$ means that for every sequence $(\mathcal U_n)_{n\geq 1}$ with $\mathcal U_n\in\mathcal A(X)$, there are choices $U_n\in\mathcal U_n$ such that \[ \{U_n:n\geq 1\}\in\mathcal B(X). \] The property $\mathsf U_{\mathrm{fin}}(\mathcal O,\Omega)$ means that for every sequence $(\mathcal U_n)_{n\geq 1}$ of members of $\mathcal O(X)$, each having no finite subcover of $X$, there exist finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal F_n:n\geq 1\right\}\in\Omega(X). \]

The record attributes to reference label [strongdiags] the identity \[ \mathsf S_1(\Omega,\gimel(\Omega)) = \mathsf U_{\mathrm{fin}}(\mathcal O,\Omega)\cap\mathsf S_1(\mathcal O,\mathcal O), \] where intersection means simultaneous satisfaction of the two properties. Using this supplied identity, the question becomes whether, for every $X$ in the intended ambient class, \[ \begin{aligned} X\models\mathsf S_1(\Omega,\Omega) \quad\Longleftrightarrow\quad &X\models\mathsf U_{\mathrm{fin}}(\mathcal O,\Omega)\\ &\text{and }X\models\mathsf S_1(\mathcal O,\mathcal O). \end{aligned} \] The identity is reproduced as a source assertion, not independently established here. It permits this classical reformulation without supplying an unverified definition of $\gimel(\Omega)$.

As background, the record also attributes to [strongdiags] the assertion that $\mathsf S_1(\Omega,\gimel(\Omega))$ is strictly stronger than $\mathsf S_1(\mathcal O,\mathcal O)$. The ambient class and hypotheses governing that strictness assertion are likewise absent. The registry labels the question open; that label is supplied curation, not an independently verified literature assessment.

Short English statement

The question asks whether selecting one member from each of a sequence of open $\omega$-covers so that the selections still form an $\omega$-cover is equivalent to having both $\mathsf U_{\mathrm{fin}}(\mathcal O,\Omega)$ and $\mathsf S_1(\mathcal O,\mathcal O)$. This reformulation uses an identity asserted in the supplied record. The intended class of spaces, cover conventions, and original meaning of $\gimel$ require source review.

Sources

• Local UnsolvedMath registry, permanent ID $800600$, exact source problem code catalogue entry; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry lists Tsaban as proposer, gives the year 2006, and labels the problem open.
• Supplied source attribution: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 28, https://arxiv.org/abs/math/0609601. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent access or verification.
• The source title and background give attribution/context label [coc8]. The displayed background identity and strict-strength assertion cite [strongdiags]. The supplied record does not expand either label into a bibliographic reference, so their authorship, hypotheses, and supporting statements remain unverified.