Selection Principles in Mathematics — Problem 40 (Tka\v{c}enko)

Definitions and mathematical statement

Under the usual interpretation of the selection-principle notation, let $K$ be a topological group with identity $e_K$. Write $\mathcal O(K)$ for the family of open covers of $K$, and put \[ \mathcal O_{\mathrm{nbd}}(K)=\left\{\{kU:k\in K\}: U\subseteq K\text{ is open and }e_K\in U\right\}. \] The property $\mathsf S_{\mathrm{fin}}(\mathcal O_{\mathrm{nbd}},\mathcal O)$ means that for every sequence $(\mathcal U_n)_{n\geq 1}$ from $\mathcal O_{\mathrm{nbd}}(K)$, there are finite subfamilies $\mathcal V_n\subseteq\mathcal U_n$ such that $\bigcup_{n\geq 1}\mathcal V_n$ covers $K$. Equivalently, for every sequence $(U_n)_{n\geq 1}$ of open identity neighborhoods, there are finite sets $F_n\subseteq K$ such that \[ K=\bigcup_{n\geq 1}F_nU_n, \qquad F_nU_n=\{fu:f\in F_n,\ u\in U_n\}. \] Problem (Tkačenko; Tsaban's Problem 40). Does ZFC prove the existence of topological groups $G$ and $H$, each satisfying $\mathsf S_{\mathrm{fin}}(\mathcal O_{\mathrm{nbd}},\mathcal O)$, whose product $G\times H$, equipped with the product topology and coordinatewise group operation, fails this property? Explicitly, the required failure means that there is a sequence $(W_n)_{n\geq 1}$ of open identity neighborhoods in $G\times H$ such that \[ \text{for every sequence of finite sets }E_n\subseteq G\times H, \qquad G\times H\ne\bigcup_{n\geq 1}E_nW_n. \] The supplied record does not define $\mathcal O_{\mathrm{nbd}}$ or specify separation axioms or further restrictions on the topological groups. The definitions above state the conventional interpretation explicitly; the intended group category requires source review.

Short English statement

Can ZFC establish the existence of two topological groups for which every sequence of identity neighborhoods admits finitely many translates at each stage covering the group altogether, while their product fails this covering property? The supplied record labels this problem open; that status is not independently verified.

Sources

• Local UnsolvedMath registry: permanent ID $800612$, exact source problem code catalogue entry, in the AMR Open Problem Lists collection. The supplied record lists Tsaban as proposer, gives the year 2006, and labels the problem open.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 40. Supplied source URL: https://arxiv.org/abs/math/0609601. The supplied background attributes the problem to Tkačenko.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance statements, not an independent source visit or verification. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.