Selection Principles in Mathematics — Problem 35 ([split])

Definitions and mathematical statement

The supplied record asks about three splitting properties but does not define its cover conventions or its ambient class of spaces. The following gives a parameterized formulation using conventional cover terminology; these missing conventions require source verification before this can be treated as a definitive reconstruction.

For a topological space $Y$, a nontrivial cover is a family $\mathcal U\subseteq\mathcal P(Y)$ such that $\bigcup\mathcal U=Y$ and $Y\notin\mathcal U$. Let $\mathcal A(Y)$ denote the intended family of admissible nontrivial covers. The supplied record does not specify whether admissible covers must be open, clopen, or countable.

An $\omega$-cover is a nontrivial cover $\mathcal U$ such that \[ \forall F\subseteq Y\;\left(F\text{ finite}\Longrightarrow\exists U\in\mathcal U\; F\subseteq U\right). \] A cover is large if every point belongs to infinitely many of its members. A $\tau$-cover is a large nontrivial cover $\mathcal U$ such that, for every $x,y\in Y$, 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. Write $\Omega_{\mathcal A}(Y)$ and $\mathrm T_{\mathcal A}(Y)$ for the admissible $\omega$-covers and admissible $\tau$-covers, respectively.

For two specified families of covers $\mathcal B(Y)$ and $\mathcal D(Y)$, use the conventional disjoint-subcover interpretation \[ \mathsf{Split}(\mathcal B,\mathcal D)(Y) \iff \forall\mathcal U\in\mathcal B(Y)\;\exists\mathcal V,\mathcal W\in\mathcal D(Y)\; \left(\mathcal V\subseteq\mathcal U\ \land\ \mathcal W\subseteq\mathcal U\ \land\ \mathcal V\cap\mathcal W=\varnothing\right). \] Here $\mathcal V$ and $\mathcal W$ are themselves covers of $Y$; this interpretation does not require $\mathcal V\cup\mathcal W=\mathcal U$. The intended definition of $\mathsf{Split}$ must also be checked against the source.

For a fixed intended class $\mathcal C$ of spaces and a fixed admissibility convention on these spaces and their powers, the question is whether any of \[ P\in\left\{ \mathsf{Split}(\Omega_{\mathcal A},\Omega_{\mathcal A}),\quad \mathsf{Split}(\Omega_{\mathcal A},\mathrm T_{\mathcal A}),\quad \mathsf{Split}(\mathrm T_{\mathcal A},\mathrm T_{\mathcal A}) \right\} \] satisfies \[ \forall X\in\mathcal C\;\left(P(X)\Longrightarrow \forall n\in\{1,2,3,\ldots\}\;P(X^n)\right), \] where $X^n$ has the product topology. Positive finite powers are used in this displayed formulation. The source record does not explicitly address the zeroth power. Neither $\mathcal C$ nor $\mathcal A$ is determined by the supplied excerpt, so the displayed question remains conditional on those choices.

Short English statement

Tsaban's Problem 35 asks whether any of the three properties $\mathsf{Split}(\Omega,\Omega)$, $\mathsf{Split}(\Omega,\mathrm T)$, and $\mathsf{Split}(\mathrm T,\mathrm T)$ passes from a space to each of its positive finite Cartesian powers. The intended spaces, admissible covers, and exact splitting convention are missing from the supplied excerpt and require verification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800607$, exact source problem code catalogue entry; upstream ID $800607$, revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 35. Supplied source URL: https://arxiv.org/abs/math/0609601. The registry records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The supplied attribution/context marker is [split]; its bibliographic expansion is absent from the record.
• The registry labels the problem open and gives an update date of 2026-10-16. This is a supplied status label, not an independently verified assessment of the mathematical literature. No separate literature-verification date or additional literature references are supplied.