Selection Principles Milestone Problems — Problem 32 ([coc5])

Definitions and mathematical statement

For families $\mathcal A,\mathcal B$ of subsets of a set $E$, the selection principle $\mathsf S_1(\mathcal A,\mathcal B)$ means that for every sequence $(A_n)_{n\geq 1}$ of members of $\mathcal A$, there are choices $a_n\in A_n$ such that $\{a_n:n\geq 1\}\in\mathcal B$. The selection principle $\mathsf S_{\mathrm{fin}}(\mathcal A,\mathcal B)$ means that for every such sequence there are finite sets $F_n\subseteq A_n$ such that $\bigcup_{n\geq 1}F_n\in\mathcal B$; the finite sets may be empty.

The supplied Problem 32 asks whether the properties \[ \mathsf S_1(\mathcal D_\Omega,\mathcal D_\Omega) \quad\text{or}\quad \mathsf S_{\mathrm{fin}}(\mathcal D_\Omega,\mathcal D_\Omega) \] are preserved under taking finite powers. The record does not define $\mathcal D_\Omega$, specify its underlying set of selectable objects, or identify the class of spaces under consideration. Consequently, a complete mathematical formulation cannot be recovered from the supplied record alone.

The preservation questions can be expressed schematically as follows. Let $\mathscr C$ denote the intended class of spaces, and let $\mathcal D_\Omega(X)$ denote the intended family associated with a space $X$. For each $\star\in\{1,\mathrm{fin}\}$, ask whether \[ \forall X\in\mathscr C\;\forall m\in\mathbb Z_{\geq 1},\qquad \mathsf S_\star\left(\mathcal D_\Omega(X),\mathcal D_\Omega(X)\right) \Longrightarrow \mathsf S_\star\left(\mathcal D_\Omega(X^m),\mathcal D_\Omega(X^m)\right). \] Here, if the intended setting is topological spaces, $X^m$ denotes the Cartesian power equipped with the product topology. This is a conditional formulation template: the meanings of $\mathscr C$ and $\mathcal D_\Omega(X)$, including any restrictions on spaces or selectable families, remain unresolved. The two values of $\star$ display the two properties mentioned in the source separately.

Short English statement

The source asks whether each of two selection properties involving $\mathcal D_\Omega$ survives positive finite powers. Their selection rules are stated above, but the supplied record omits the definition of $\mathcal D_\Omega$ and the intended class of spaces. Those omissions require clarification before this can be a complete mathematical statement.

Sources

• Local UnsolvedMath registry: permanent ID $800659$, exact source problem code catalogue entry; title: Selection Principles Milestone Problems --- Problem 32 ([coc5]). The supplied registry attributes the record to Tsaban, with proposed year 2003, and labels its status as open. This is a supplied status label, not an independently verified mathematical or literature assessment.
• Supplied source citation: Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 32, 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 independent source access.
• Supplied attribution/context label: [coc5]. Its bibliographic referent and any qualifications it supplies are absent from the record.