Selection Principles Milestone Problems — Conjecture 24 ([splittability, Conjecture 8.7])

Definitions and mathematical statement

The supplied record states the following conjecture, attributed to Tsaban (2003):

Neither of the classes $\mathsf{Split}(\mathrm{T},\mathrm{T})$ and $\binom{\mathrm{T}}{\Gamma}$ is provably closed under taking finite products.

This is a metamathematical assertion. The supplied excerpt does not specify the underlying axiom system, the ambient class of spaces, or the admissibility conventions for covers. These omissions prevent an unconditional precise formulation. The following gives an explicit formulation template, with those parameters left unresolved.

For a topological space $X$, let $\mathcal C(X)$ denote the intended collection of admissible covers of $X$. Its restrictions, including openness, countability, and exclusion of a member equal to $X$, must be recovered from the source. For a cover $\mathcal U$ of $X$, set \[ \mathcal U(x)=\{U\in\mathcal U:x\in U\}\qquad(x\in X). \] Under the usual cover-theoretic interpretation of the symbols, a $\tau$-cover is a cover $\mathcal U$ such that every $\mathcal U(x)$ is infinite and, for all $x,y\in X$, \[ \mathcal U(x)\setminus\mathcal U(y)\text{ is finite} \quad\text{or}\quad \mathcal U(y)\setminus\mathcal U(x)\text{ is finite}. \] A $\gamma$-cover is an infinite cover $\mathcal U$ such that \[ \{U\in\mathcal U:x\notin U\}\text{ is finite for every }x\in X. \] Write $\mathrm T(X)$ and $\Gamma(X)$ for the admissible covers satisfying these respective conditions. These definitions explain the conventional notation; their applicability and the admissibility conventions have not been verified from the supplied excerpt.

With these conventions, define \[ \begin{aligned} X\in\mathsf{Split}(\mathrm T,\mathrm T) \quad\Longleftrightarrow\quad& \text{for every }\mathcal U\in\mathrm T(X),\text{ there are disjoint }\mathcal U_0,\mathcal U_1\in\mathrm T(X)\\ &\text{such that }\mathcal U=\mathcal U_0\cup\mathcal U_1, \end{aligned} \] and \[ X\in\binom{\mathrm T}{\Gamma} \quad\Longleftrightarrow\quad \forall\mathcal U\in\mathrm T(X)\ \exists\mathcal V\subseteq\mathcal U\ \left(\mathcal V\in\Gamma(X)\right). \] The splitting convention should also be checked against the original source.

Let $\mathscr K$ denote the intended ambient class of spaces. For either property $P$, its finite-product closure assertion is \[ \operatorname{FP}_{\mathscr K}(P):\quad \forall n\in{\mathbb{N}},\ n\geq1,\quad \forall X_1,\ldots,X_n\in\mathscr K,\quad \left(\bigwedge_{i=1}^{n}P(X_i)\right) \Longrightarrow P\!\left(\prod_{i=1}^{n}X_i\right), \] where the product carries the product topology. If $B$ is the intended background axiom system, the supplied conjecture has the form \[ B\nvdash\operatorname{FP}_{\mathscr K}\left(\mathsf{Split}(\mathrm T,\mathrm T)\right) \qquad\text{and}\qquad B\nvdash\operatorname{FP}_{\mathscr K}\!\left(\binom{\mathrm T}{\Gamma}\right). \] Neither $B$, $\mathscr K$, nor $\mathcal C(X)$ is determined by the supplied excerpt. The convention concerning the empty product is also unspecified. In particular, the conjecture must not be replaced by an assertion that counterexamples exist in the actual universe, or by an assertion of independence in both directions.

Short English statement

Tsaban's supplied Conjecture 24 says that finite-product closure is not provable for either $\mathsf{Split}(\mathrm T,\mathrm T)$ or $\binom{\mathrm T}{\Gamma}$. The former concerns splitting every admissible $\tau$-cover into two $\tau$-covers; the latter concerns finding a $\gamma$-subcover in every admissible $\tau$-cover. The axiom system, ambient spaces, and cover conventions require source verification.

Sources

• Local UnsolvedMath registry, permanent ID $800651$, source problem code catalogue entry; title: Selection Principles Milestone Problems --- Conjecture 24 ([splittability, Conjecture 8.7]). The record attributes the conjecture to Tsaban, proposed in 2003, and supplies the status label open. This label is supplied registry metadata, not an independently verified mathematical or literature assessment.
• Supplied source: Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Conjecture 24, https://arxiv.org/abs/math/0312182. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of access or verification for this draft.
• Supplied attribution/context: [splittability, Conjecture 8.7]. The record does not provide the bibliographic entry corresponding to splittability; its authorship and publication details remain unresolved.