Selection Principles Milestone Problems — Problem 6 ([CBC, Problem 50])

Definitions and mathematical statement

Let $X$ be a topological space. A countable Borel cover of $X$ is a countable family $\mathcal U$ of Borel subsets of $X$ whose union is $X$, with $X\notin\mathcal U$. Such a family is an $\omega$-cover if every finite subset of $X$ is contained in some member of $\mathcal U$. It is a $\gamma$-cover if it is infinite and, for every $x\in X$, the set $\{U\in\mathcal U:x\notin U\}$ is finite. Write $\mathcal B_\Omega(X)$ and $\mathcal B_\Gamma(X)$ for the families of countable Borel $\omega$-covers and countable Borel $\gamma$-covers, respectively.

For families $\mathcal A(X)$ and $\mathcal D(X)$ of covers, the property $\mathsf S_1(\mathcal A,\mathcal D)$ means that for every sequence $(\mathcal U_n)_{n\geq1}$ with $\mathcal U_n\in\mathcal A(X)$, there are choices $U_n\in\mathcal U_n$ such that $\{U_n:n\geq1\}\in\mathcal D(X)$. The property $\mathsf S_{\mathrm{fin}}(\mathcal A,\mathcal D)$ means that for every such sequence there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$, possibly empty, such that $\bigcup_{n\geq1}\mathcal F_n\in\mathcal D(X)$. These selected collections are families of sets; in particular, a selected $\gamma$-cover must have infinitely many distinct members.

Consider the three properties \[ P_1=\mathsf S_1(\mathcal B_\Omega,\mathcal B_\Gamma),\qquad P_2=\mathsf S_1(\mathcal B_\Omega,\mathcal B_\Omega),\qquad P_3=\mathsf S_{\mathrm{fin}}(\mathcal B_\Omega,\mathcal B_\Omega). \] For a specified ambient class $\mathscr C$ of spaces, the finite-power closure question for each $i\in\{1,2,3\}$ is whether \[ \forall X\in\mathscr C\;\forall m\in{\mathbb{Z}}_{\geq1},\qquad P_i(X)\Longrightarrow P_i(X^m), \] where $X^m$ has the product topology and its cover families are defined using Borel subsets of that product.

Problem. Determine which, if any, of these three classes is closed under taking finite powers in the intended ambient category. The supplied record does not specify that category or reproduce the source's cover conventions. The definitions above give the usual countable Borel-cover interpretation, subject to source verification; the displayed closure questions remain conditional on identifying $\mathscr C$. In particular, this draft does not silently identify the intended category with all topological spaces or with any particular class of subspaces of ${\mathbb{R}}$.

Short English statement

For each of three Borel-cover selection properties, ask whether every positive finite Cartesian power of a space with that property also has the property. The intended class of spaces is missing from the supplied excerpt and requires verification. The registry labels the problem open; that label is supplied metadata, not an independently verified assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800634$, source code catalogue entry, title: Selection Principles Milestone Problems --- Problem 6 ([CBC, Problem 50]). The record lists Tsaban as proposer and $2003$ as the proposed year.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 6. Supplied source URL: https://arxiv.org/abs/math/0312182. The supplied provenance records access on $2026\text{-}10\text{-}16$ and extraction method arxiv_latex; these are source-record provenance claims, not access or verification performed for this draft.
• Supplied attribution/context: [CBC, Problem 50]. The record does not supply the bibliographic expansion of CBC, so the underlying reference and original attribution remain unresolved.
• The registry supplies status open and upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. No independent literature assessment is asserted here.