Selection Principles in Mathematics — Problem 5 ([SFH])

Definitions and mathematical statement

Regard a set of reals $M\subseteq{\mathbb{R}}$ as a space with the usual subspace topology, and put ${\mathbb{N}}=\{1,2,\ldots\}$. The supplied excerpt does not define its cover conventions; the following gives the standard topological interpretation, subject to confirmation against the source.

An open cover of $M$ is a family $\mathcal U$ of open subsets of $M$ such that $\bigcup\mathcal U=M$ and $M\notin\mathcal U$. Write $\mathcal O(M)$ for the class of these covers. An $\omega$-cover is an open cover $\mathcal U$ such that every finite subset of $M$ is contained in some member of $\mathcal U$; write $\Omega(M)$ for this class. A $\gamma$-cover is an infinite open cover $\mathcal U$ such that, for every $x\in M$, the set $\{U\in\mathcal U:x\notin U\}$ is finite; write $\Gamma(M)$ for this class. These definitions impose no countability restriction on individual covers.

The property $\mathsf S_{\mathrm{fin}}(\Omega,\Omega)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ in $\Omega(M)$, there are finite subfamilies $\mathcal V_n\subseteq\mathcal U_n$ such that \[ \bigcup_{n\in{\mathbb{N}}}\mathcal V_n\in\Omega(M). \] The property $\mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ in $\mathcal O(M)$, each having no finite subcover of $M$, there are finite subfamilies $\mathcal V_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal V_n:n\in{\mathbb{N}}\right\}\in\Gamma(M). \] Thus failure of this latter property means that some such sequence admits no sequence of finite selections producing a $\gamma$-cover in this way.

The source asks: Is there a direct, non-dichotomic construction of a set $M\subseteq{\mathbb{R}}$ satisfying \[ \mathsf S_{\mathrm{fin}}(\Omega,\Omega) \quad\text{and}\quad \neg\mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma)? \] The requested construction method is part of the question. The supplied record does not define "direct" or "non-dichotomic", identify the dichotomy to be avoided, or specify any additional set-theoretic hypotheses. Consequently, the displayed properties specify the requested set under the conventions above, but do not fully formalize the methodological requirement.

Short English statement

Find a direct, non-dichotomic construction of a set of reals with $\mathsf S_{\mathrm{fin}}(\Omega,\Omega)$ but without $\mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma)$. The meaning of the construction requirement and the source's exact cover conventions require confirmation. The registry supplies an open label; no independent status verification is asserted.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800579$, source problem code catalogue entry, title "Selection Principles in Mathematics --- Problem 5 ([SFH])". The record lists Tsaban as proposer and $2006$ as the proposed year; its supplied status is open.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 5. Supplied source URL: https://arxiv.org/abs/math/0609601. The record reports extraction by arxiv_latex and an access date of 2026-10-16; these are supplied provenance details, not a claim of independent source access or verification.
• The supplied background gives attribution/context as [SFH]; its expansion and bibliographic referent are not supplied.