Selection Principles Milestone Problems — Problem 1

Definitions and mathematical statement

The supplied record leaves the class of spaces and the cover conventions unspecified. The following is a conditional formulation using standard topological meanings of the notation; the intended class and conventions require source verification.

Fix a class $\mathcal C$ of topological spaces. For each $X\in\mathcal C$, fix a collection $\mathscr O(X)$ of admissible open covers of $X$, with $X\notin\mathcal U$ for every $\mathcal U\in\mathscr O(X)$. An open cover is a family of open subsets whose union is $X$. Whether admissible covers must be countable is unresolved in the supplied record. Define \[ \Omega(X)=\left\{\mathcal U\in\mathscr O(X): (\forall F\subseteq X\text{ finite})(\exists U\in\mathcal U)\ F\subseteq U\right\} \] and \[ \Gamma(X)=\left\{\mathcal U\in\mathscr O(X):\mathcal U\text{ is infinite and }(\forall x\in X)\ \{U\in\mathcal U:x\notin U\}\text{ is finite}\right\}. \] For classes $\mathcal A(X),\mathcal B(X)$ of covers, the property $\mathsf S_{\mathrm{fin}}(\mathcal A,\mathcal B)$ means that for every sequence $(\mathcal U_n)_{n\geq1}$ in $\mathcal A(X)$, there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \bigcup_{n\geq1}\mathcal F_n\in\mathcal B(X). \] The property $\mathsf U_{\mathrm{fin}}(\mathcal A,\mathcal B)$ means that for every sequence $(\mathcal U_n)_{n\geq1}$ in $\mathcal A(X)$, each $\mathcal U_n$ having no finite subcover of $X$, there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal F_n:n\geq1\right\}\in\mathcal B(X). \] Here finite selections may be empty, and $\bigcup\varnothing=\varnothing$; this convention also requires verification against the original source.

With the intended class $\mathcal C$ and cover conventions fixed, Problem 1 asks whether \[ (\forall X\in\mathcal C)\quad X\models\mathsf U_{\mathrm{fin}}(\Gamma,\Omega) \ \Longleftrightarrow\ X\models\mathsf S_{\mathrm{fin}}(\Gamma,\Omega). \] If this equivalence fails, it further asks whether \[ (\forall X\in\mathcal C)\quad X\models\mathsf U_{\mathrm{fin}}(\Gamma,\Gamma) \ \Longrightarrow\ X\models\mathsf S_{\mathrm{fin}}(\Gamma,\Omega). \] The unspecified class of spaces and cover conventions prevent identifying this conditional formulation with a uniquely determined source statement. The registry labels the problem open; this is supplied status information, not an independently verified mathematical or literature assessment.

Short English statement

For the intended spaces and cover conventions, ask whether finite selections from sequences of $\gamma$-covers give equivalent properties when their individual members or their unions must form an $\omega$-cover. If the properties differ, ask whether the corresponding union-selection property with a $\gamma$-cover as output implies $\mathsf S_{\mathrm{fin}}(\Gamma,\Omega)$. The intended domain and conventions remain to be verified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800629$, exact source problem code catalogue entry, title "Selection Principles Milestone Problems --- Problem 1." The record attributes the problem to Tsaban, with proposed year 2003, and supplies status "open."
• Supplied source list: Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 1, https://arxiv.org/abs/math/0312182. The record reports extraction from arXiv LaTeX and access on 2026-10-16; these are supplied provenance details, not independently verified access or a current literature assessment.
• The supplied statement includes the reference label "coc2, Problems 1 and 2." The corresponding bibliography entry is absent from the supplied record, so its authorship and publication details remain unresolved.