Selection Principles Milestone Problems — Problem 7 ([CBC, Problems 19 and 21])

Definitions and mathematical statement

For a topological space $Y$, a Borel cover is a family of Borel subsets of $Y$ whose union is $Y$. For this draft, adopt the standard convention that covers are countable and do not contain $Y$ itself. Write $\mathcal B(Y)$ for these covers. An $\omega$-cover is a cover $\mathcal U$ such that every finite subset of $Y$ is contained in some member of $\mathcal U$. Write $\mathcal B_\Omega(Y)$ for the Borel $\omega$-covers. A $\gamma$-cover is an infinite cover $\mathcal U$ such that, for every $y\in Y$, the set $\{U\in\mathcal U:y\notin U\}$ is finite. Write $\mathcal B_\Gamma(Y)$ for the Borel $\gamma$-covers.

For classes $\mathcal A$ and $\mathcal C$ of covers of the same space, $\mathsf S_1(\mathcal A,\mathcal C)$ means that for every sequence $(\mathcal U_n)_{n\geq 1}$ with $\mathcal U_n\in\mathcal A$, there are choices $U_n\in\mathcal U_n$ such that $\{U_n:n\geq 1\}\in\mathcal C$. The property $\mathsf S_{\mathrm{fin}}(\mathcal A,\mathcal C)$ means that for every such sequence there are finite, possibly empty, subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that $\bigcup_{n\geq 1}\mathcal F_n\in\mathcal C$.

The supplied Problem 7 asks whether each of the following implications holds for every $X$ in the intended class of spaces: \[ \mathsf S_1\left(\mathcal B_\Omega(X),\mathcal B_\Omega(X)\right) \quad\Longrightarrow\quad \forall k\geq 1\;\mathsf S_1\left(\mathcal B(X^k),\mathcal B(X^k)\right), \] \[ \mathsf S_{\mathrm{fin}}\left(\mathcal B_\Omega(X),\mathcal B_\Omega(X)\right) \quad\Longrightarrow\quad \forall k\geq 1\;\mathsf S_1\left(\mathcal B_\Gamma(X^k),\mathcal B(X^k)\right). \] Here $k$ ranges over positive integers, and $X^k$ carries the product topology; Borel sets are taken in that topology.

Unresolved scope and conventions. The supplied record does not specify the intended class of spaces $X$, nor does it supply definitions or cover conventions. The definitions above provide a standard working interpretation requiring confirmation against the source context. In particular, these questions are not asserted here to have been posed for all topological spaces.

Short English statement

The two questions ask whether the stated Borel selection properties of $X$ force the respective Borel selection properties in every positive finite power of $X$. The intended class of spaces and the cover conventions require source confirmation. The registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $800635$, exact source problem code catalogue entry; supplied title: Selection Principles Milestone Problems --- Problem 7 ([CBC, Problems 19 and 21]). Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 7. Supplied source URL: https://arxiv.org/abs/math/0312182. The registry attributes the problem to Tsaban and records proposed year 2003.
• The supplied attribution/context is explicitly [CBC, Problems 19 and 21]. No bibliographic expansion of CBC is supplied.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims, not a claim of source access or independent verification by this draft. The supplied registry status is open.