Selection Principles in Mathematics — Problem 32 ([AddQuad])

Definitions and mathematical statement

The supplied question concerns the lower bound \[ \mathfrak{u}\leq \operatorname{add}\left(\mathsf{Split}(\mathrm{T},\mathrm{T})\right). \] This inequality is reported by the source record; it is not proved or independently verified here.

The ultrafilter number $\mathfrak{u}$ is the least cardinality of a base for a free ultrafilter on ${\mathbb{N}}$. Explicitly, writing $B\subseteq^* A$ when $B\setminus A$ is finite, \[ \mathfrak{u}=\min\left\{|\mathcal{B}|:\begin{array}{l} \mathcal{U}\text{ is a free ultrafilter on }{\mathbb{N}},\quad \mathcal{B}\subseteq\mathcal{U},\\ \text{and }(\forall A\in\mathcal{U})(\exists B\in\mathcal{B})\ B\subseteq^* A \end{array}\right\}. \] An ultrafilter is free if it contains no finite set.

The remaining notation requires conventions absent from the supplied record. Under a usual nontrivial-cover convention, a family $\mathcal{V}$ of subsets of a space $X$ is a $\tau$-cover if $X\notin\mathcal{V}$, every point belongs to infinitely many members of $\mathcal{V}$, and, for every $x,y\in X$, at least one of \[ \{V\in\mathcal{V}:x\in V,\ y\notin V\}, \qquad \{V\in\mathcal{V}:y\in V,\ x\notin V\} \] is finite. The supplied record does not specify the admissible spaces or the openness, countability, or other restrictions defining its class $\mathrm{T}(X)$; the preceding description therefore supplies a conventional combinatorial interpretation, not a verified identification of all the source's conventions.

For a specified class $\mathrm{T}(X)$, the splitting property is expressed by \[ X\models\mathsf{Split}(\mathrm{T},\mathrm{T}) \quad\Longleftrightarrow\quad (\forall\mathcal{V}\in\mathrm{T}(X)) (\exists\mathcal{V}_0,\mathcal{V}_1\in\mathrm{T}(X))\ \left[\mathcal{V}=\mathcal{V}_0\mathbin{\dot\cup}\mathcal{V}_1\right]. \] Once an ambient space $Z$ and the cover conventions have been fixed, let \[ \mathcal{P}_Z=\{X\subseteq Z:X\models\mathsf{Split}(\mathrm{T},\mathrm{T})\}, \] with the subspace topology on each $X$. The usual union additivity number is then \[ \operatorname{add}(\mathcal{P}_Z)= \min\left\{\kappa:\exists(X_\alpha)_{\alpha<\kappa}\in\mathcal{P}_Z^\kappa, \ \bigcup_{\alpha<\kappa}X_\alpha\notin\mathcal{P}_Z\right\}, \] provided a witnessing family exists. The record does not specify the intended ambient space or the convention when no witness exists.

Source question (Problem 32). Can the reported lower bound $\mathfrak{u}$ on $\operatorname{add}(\mathsf{Split}(\mathrm{T},\mathrm{T}))$ be improved?

The record names neither a proposed stronger bound nor the assumptions under which an improvement is sought. In particular, it does not explicitly ask for the strict inequality $\mathfrak{u}<\operatorname{add}(\mathsf{Split}(\mathrm{T},\mathrm{T}))$. A fully specified conjecture requires clarification of these points and of the cover and ambient-space conventions.

Short English statement

Tsaban's Problem 32 asks whether the reported ultrafilter-number lower bound for the additivity of $\mathsf{Split}(\mathrm{T},\mathrm{T})$ can be strengthened. The supplied record leaves the precise cover conventions, ambient class of spaces, and intended stronger bound unspecified. Its open label is supplied registry status, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800604$, source problem code catalogue entry; title: Selection Principles in Mathematics --- Problem 32 ([AddQuad]). The record attributes the problem to Tsaban, with proposed year 2006, and labels it open.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 32. 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 evidence of an independent source visit here.
• Supplied attribution/context label: [AddQuad]. No expansion or bibliographic identification of this label is included in the complete record.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry's update timestamp is 2026-10-16T14:00:00Z; it is not an independently verified literature assessment.