Selection Principles Milestone Problems — Problem 8 ([AddQuad])

Definitions and mathematical statement

Let $T$ be an ambient topological space and give each $X\subseteq T$ its subspace topology. The ambient space, permitted subspaces, and cover conventions must be specified to obtain a definite formulation; the supplied record does not specify them.

Under the usual open-cover interpretation, an admissible cover $\mathcal U$ of $X$ consists of open subsets of $X$, satisfies $\bigcup\mathcal U=X$, and does not contain $X$ itself. An $\omega$-cover is such a cover satisfying \[ \forall F\subseteq X\;\left(F\text{ finite}\Longrightarrow \exists U\in\mathcal U\;(F\subseteq U)\right). \] A $\gamma$-cover is an infinite such cover satisfying \[ \forall x\in X,\qquad \{U\in\mathcal U:x\notin U\}\text{ is finite}. \] Write $\Omega(X)$ and $\Gamma(X)$ for the corresponding families of admissible covers. Whether admissible covers must be countable is unresolved in the supplied record; these definitions are to be restricted accordingly once that convention is established.

For families of covers $\mathcal A(X)$ and $\mathcal B(X)$, the property $\mathsf S_1(\mathcal A,\mathcal B)$ means that for every sequence $(\mathcal U_n)_{n\geq1}$ from $\mathcal A(X)$, there are choices $U_n\in\mathcal U_n$ such that $\{U_n:n\geq1\}\in\mathcal B(X)$. The property $\mathsf S_{\mathrm{fin}}(\mathcal A,\mathcal B)$ means that for every such sequence there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \bigcup_{n\geq1}\mathcal F_n\in\mathcal B(X). \] Here the displayed union is a family of open sets.

Question (source Problem 8, [AddQuad]). Is at least one of the three properties \[ \mathsf S_{\mathrm{fin}}(\Omega,\Omega),\qquad \mathsf S_1(\Gamma,\Omega),\qquad \mathsf S_{\mathrm{fin}}(\Gamma,\Omega) \] consistently closed under finite unions? For each candidate property $P$, the finite-union assertion, once the intended ambient setting is fixed, is \[ \forall m\geq1\;\forall X_1,\ldots,X_m\subseteq T,\qquad \left(\bigwedge_{j=1}^{m}P(X_j)\right) \Longrightarrow P\!\left(\bigcup_{j=1}^{m}X_j\right), \] with the quantifiers restricted to the intended permitted subspaces. Under the usual set-theoretic reading, the question asks whether this assertion for at least one candidate is consistent with $\mathrm{ZFC}$. Neither that base theory nor the intended ambient setting is explicit in the supplied record. Thus this is a formulation schema with unresolved qualifications, not a uniquely specified consistency assertion.

Short English statement

The source asks whether any of three selection properties can consistently be preserved when finitely many spaces having that property are united. A precise interpretation still requires the intended class of spaces, cover countability conventions, and consistency base theory. The registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $800636$, exact source problem code catalogue entry; supplied title: Selection Principles Milestone Problems --- Problem 8 ([AddQuad]). The record attributes the problem to Tsaban, with proposed year $2003$, and supplies status open.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 8. Supplied source URL: https://arxiv.org/abs/math/0312182. The record reports extraction by arxiv_latex and access on 2026-10-16; these are supplied provenance details, not an independently verified visit or literature assessment.
• The supplied attribution/context marker is [AddQuad]. Its bibliographic expansion is not included in the record.