Selection Principles in Mathematics — Problem 30 ([split])

Definitions and mathematical statement

The supplied record does not specify the ambient class of spaces, admissible covers, or the precise splitting convention. The following is a conditional reconstruction using standard cover terminology; these conventions require comparison with the cited source.

Let $\mathcal D$ denote the intended class of topological spaces. For each $X\in\mathcal D$, let $\mathcal C(X)$ be the intended collection of open covers of $X$ whose members are proper subsets of $X$. Whether these covers must be countable is unresolved in the supplied excerpt. Define \[ \begin{aligned} \Lambda(X)&=\{\mathcal U\in\mathcal C(X):\text{each }x\in X\text{ belongs to infinitely many }U\in\mathcal U\},\\ \Omega(X)&=\{\mathcal U\in\mathcal C(X):\text{each finite }F\subseteq X\text{ is contained in some }U\in\mathcal U\},\\ \Gamma(X)&=\{\mathcal U\in\mathcal C(X):\mathcal U\text{ is infinite and each }x\in X\text{ belongs to all but finitely many }U\in\mathcal U\}. \end{aligned} \] A cover $\mathcal U\in\Lambda(X)$ belongs to $\mathrm T(X)$ if, for every $x,y\in X$, at least one of the sets \[ \{U\in\mathcal U:x\in U,\ y\notin U\},\qquad \{U\in\mathcal U:y\in U,\ x\notin U\} \] is finite.

For this reconstruction, $X\models\mathsf{Split}(\mathcal A,\mathcal B)$ means \[ \forall\mathcal U\in\mathcal A(X)\ \exists\mathcal V_0,\mathcal V_1\subseteq\mathcal U: \quad \mathcal V_0\cap\mathcal V_1=\varnothing, \qquad \mathcal V_0,\mathcal V_1\in\mathcal B(X). \] Here the two subcovers need not exhaust $\mathcal U$. Equivalently, $\mathcal U$ can be partitioned into two families each containing a member of $\mathcal B(X)$. The excerpt does not establish whether this convention, or the potentially stronger requirement that both parts themselves belong to $\mathcal B(X)$, is intended.

With these qualifications, Problem 30 asks whether \[ \forall X\in\mathcal D,\qquad X\models\mathsf{Split}(\Omega,\mathrm T) \ \Longrightarrow\ X\models\mathsf{Split}(\mathrm T,\Gamma). \tag{1} \] If (1) is false, it asks whether nevertheless \[ \forall X\in\mathcal D,\qquad X\models\mathsf{Split}(\Omega,\mathrm T) \ \Longrightarrow\ X\models\mathsf{Split}(\Lambda,\Lambda). \tag{2} \] The intended choices of $\mathcal D$, $\mathcal C(X)$, and splitting convention must be recovered before this becomes a fully specified formulation of the source problem.

Short English statement

Does the property of splitting every $\omega$-cover into two disjoint $\tau$-subcovers imply the corresponding splitting property from $\tau$-covers to $\gamma$-covers? If that implication fails, does it still imply that every large cover has two disjoint large subcovers? This reading is conditional on unresolved ambient-space and cover conventions.

Sources

• Local UnsolvedMath registry, permanent ID $800602$, exact source problem code catalogue entry, supplied complete source record. The registry attributes the problem to Tsaban, gives proposed year 2006, and labels it open; this status is supplied metadata, not an independently verified mathematical assessment.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 30. Supplied source URL: https://arxiv.org/abs/math/0609601. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independent source verification.
• The supplied attribution/context field contains [split]. No bibliographic expansion of this reference label is supplied, so its intended reference and any additional attribution remain unresolved.