Selection Principles in Mathematics — Problem 2

Definitions and mathematical statement

Give every subset of ${\mathbb{R}}$ its subspace topology. A countable open cover of $X$ is a countable family of subsets open in $X$ whose union is $X$. A $\gamma$-cover of $X$ is an infinite open cover such that each point of $X$ belongs to all but finitely many members. Write $\mathcal O$ for the family of countable open covers of $X$, and $\Gamma$ for the family of countable $\gamma$-covers of $X$. The selection property $\mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma)$, also called the Hurewicz property, means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ of members of $\mathcal O$, none of which has a finite subcover, there are finite subfamilies $\mathcal V_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal V_n:n\in{\mathbb{N}}\right\}\in\Gamma. \] A set $F\subseteq{\mathbb{R}}$ is $\sigma$-compact if $F=\bigcup_{n\in{\mathbb{N}}}K_n$ for compact sets $K_n\subseteq{\mathbb{R}}$. A set $B\subseteq{\mathbb{R}}$ is Borel if it belongs to the $\sigma$-algebra generated by the open subsets of ${\mathbb{R}}$. It is $F_{\sigma\delta}$ if \[ B=\bigcap_{m\in{\mathbb{N}}}\bigcup_{n\in{\mathbb{N}}}C_{m,n} \] for closed sets $C_{m,n}\subseteq{\mathbb{R}}$.

Under the non-strict inclusion reading of the source notation, the first question is whether \[ \begin{gathered} \forall B\subseteq{\mathbb{R}}\ \forall X\subseteq B,\\ \bigl[B\text{ is Borel and }X\text{ satisfies }\mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma)\bigr]\\ \Longrightarrow\ \exists F\subseteq{\mathbb{R}}\, \bigl[F\text{ is }\sigma\text{-compact and }X\subseteq F\subseteq B\bigr]. \end{gathered} \] The second question asks whether the same assertion holds when the hypothesis that $B$ is Borel is replaced by the stronger hypothesis that $B$ is $F_{\sigma\delta}$.

The source writes $X\subset F\subset B$ in its conclusion without specifying whether either containment must be proper; the displayed formulation uses non-strict containment and leaves that notational qualification for review. The source also attaches the unresolved reference label BZSepar, qualified as its preprint version. The supplied registry labels the problem open; this is a supplied status label, not an independently verified assessment.

Short English statement

Must every Hurewicz subset $X$ of a Borel set $B\subseteq{\mathbb{R}}$ be contained in a $\sigma$-compact set lying inside $B$? The source asks the same question when $B$ is $F_{\sigma\delta}$. Properness of the intermediate containments is not specified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800576$, exact source problem code catalogue entry, title "Selection Principles in Mathematics --- Problem 2," in the AMR Open Problem Lists collection. The record attributes the problem to Tsaban, with proposed year $2006$, and supplies the status label "open."
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 2. Supplied source URL: https://arxiv.org/abs/math/0609601. 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.
• The supplied statement includes the reference label BZSepar, explicitly qualified as "preprint version." No bibliographic identification or separate link for that reference is supplied.