Selection Principles Milestone Problems — Problem 17

Definitions and mathematical statement

The supplied record does not specify the ambient class of spaces or define its starred groupability notation. Consequently, the question below is a conditional formulation requiring those conventions to be recovered from the source.

For a topological space $X$, the standard unstarred cover notation can be described as follows. An open cover $\mathcal U$ consists of open subsets of $X$ whose union is $X$; under the usual nontrivial-cover convention, $X\notin\mathcal U$. Write $\mathcal O(X)$ for these covers. An $\omega$-cover is a member $\mathcal U\in\mathcal O(X)$ such that every finite subset of $X$ is contained in some member of $\mathcal U$; write $\Omega(X)$ for this class. A $\gamma$-cover is an infinite member $\mathcal U\in\mathcal O(X)$ such that each point of $X$ belongs to all but finitely many members; write $\Gamma(X)$ for this class. A $\tau$-cover is a member $\mathcal U\in\mathcal O(X)$ such that each point belongs to infinitely many members and, for every $x,y\in X$, at least one of \[ \{U\in\mathcal U:x\in U,\ y\notin U\},\qquad \{U\in\mathcal U:y\in U,\ x\notin U\} \] is finite. Write $\mathrm T(X)$ for this class. These descriptions explain standard notation; the supplied record does not confirm its cover conventions, including any countability restriction.

For classes $\mathcal A(X)$ and $\mathcal B(X)$ of covers, $\mathsf S_{\mathrm{fin}}(\mathcal A,\mathcal B)$ means that for every sequence $(\mathcal U_n)_{n\geq1}$ from $\mathcal A(X)$, there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \bigcup_{n\geq1}\mathcal F_n\in\mathcal B(X). \] The property $\mathsf U_{\mathrm{fin}}(\mathcal A,\mathcal B)$ has the same selection quantifiers restricted to sequences for which no $\mathcal U_n$ has a finite subcover, and requires instead \[ \left\{\bigcup\mathcal F_n:n\geq1\right\}\in\mathcal B(X). \]

For $\alpha\in\{\gamma,\omega,\tau\}$, let $\mathscr G_\alpha(X)$ stand for the source's family denoted $\mathcal O^{\alpha\text{-*gp*}}$. This is a placeholder for missing source definitions, not a definition of groupability. In particular, the supplied record does not explain the stars or the required grouping conditions.

Question (Tsaban, supplied Problem 17). Let $\mathscr X$ denote the intended, unspecified class of spaces. With the source's cover and groupability conventions restored, is it true that \[ \forall X\in\mathscr X,\qquad X\models\mathsf U_{\mathrm{fin}}(\Gamma,\mathrm T) \quad\Longleftrightarrow\quad X\models\mathsf S_{\mathrm{fin}}(\Omega,\mathscr G_\tau)? \] Here $X\models P$ means that $X$ has selection property $P$.

The supplied context recalls the equivalence \[ \mathsf U_{\mathrm{fin}}(\Gamma,\mathcal O) \Longleftrightarrow \mathsf S_{\mathrm{fin}}(\Omega,\mathcal O), \] and reports that source reference [coc7] proves \[ \mathsf U_{\mathrm{fin}}(\Gamma,\Gamma) \Longleftrightarrow \mathsf S_{\mathrm{fin}}(\Omega,\mathscr G_\gamma), \] while source reference [coc8] proves \[ \mathsf U_{\mathrm{fin}}(\Gamma,\Omega) \Longleftrightarrow \mathsf S_{\mathrm{fin}}(\Omega,\mathscr G_\omega). \] These are supplied contextual assertions with unresolved source conventions, not independently verified results here. The registry labels the question open; that label is supplied metadata rather than a verified current literature assessment.

Short English statement

Tsaban's supplied Problem 17 asks whether $\mathsf U_{\mathrm{fin}}(\Gamma,\mathrm T)$ is equivalent to selecting finitely many sets from each $\omega$-cover so that the selected family has the source's $\tau$-groupability property. The intended spaces and the starred groupability definition are missing, so the formulation remains conditional and requires human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800644$, exact source problem code catalogue entry, collection AMR Open Problem Lists; supplied title: Selection Principles Milestone Problems --- Problem 17. The record attributes the problem to Tsaban, with proposed year 2003, and labels it open.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 17. Supplied source URL: https://arxiv.org/abs/math/0312182. The record states access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not an independent source visit or verification here.
• Source-local reference [coc7] is credited in the supplied statement with the equivalence involving $\mathsf U_{\mathrm{fin}}(\Gamma,\Gamma)$. Its bibliographic entry is not supplied.
• Source-local reference [coc8] is credited in the supplied statement with the equivalence involving $\mathsf U_{\mathrm{fin}}(\Gamma,\Omega)$. Its bibliographic entry is not supplied.