Selection Principles Milestone Problems — Problem 28 ([GTpseq, Problem 3.14(2)])

Definitions and mathematical statement

For families $\mathcal A$ and $\mathcal B$ of covers of a space $X$, define \[ \binom{\mathcal A}{\mathcal B} \quad\Longleftrightarrow\quad (\forall\mathcal U\in\mathcal A)(\exists\mathcal V\subseteq\mathcal U)\;\mathcal V\in\mathcal B. \] In standard selection-principle terminology, an open $\omega$-cover of $X$ is a family $\mathcal U$ of open subsets of $X$ such that $X\notin\mathcal U$ and every finite subset of $X$ is contained in some member of $\mathcal U$. An open $\gamma$-cover is an infinite open cover $\mathcal U$ such that each point of $X$ belongs to all but finitely many members of $\mathcal U$. The symbols $\Omega$ and $\Gamma$ conventionally denote the corresponding families. The supplied record does not specify countability restrictions or the full cover conventions, so their exact source-specific meanings require confirmation.

Let $\mathscr P$ stand for the intended collection of ultrafilters. The supplied record does not specify their underlying set, whether they must be free, or the definition of the associated cover family $\mathcal O_p$. With these unresolved source parameters retained, the question is whether every space $X$ in the intended class satisfies \[ \left[(\forall p\in\mathscr P)\;\binom{\Omega(X)}{\mathcal O_p(X)}\right] \quad\Longrightarrow\quad \binom{\Omega(X)}{\Gamma(X)}. \] Equivalently, the hypothesis says \[ (\forall p\in\mathscr P)(\forall\mathcal U\in\Omega(X)) (\exists\mathcal V\subseteq\mathcal U)\;\mathcal V\in\mathcal O_p(X), \] and the requested conclusion is \[ (\forall\mathcal U\in\Omega(X)) (\exists\mathcal V\subseteq\mathcal U)\;\mathcal V\in\Gamma(X). \] Thus the selected subcover in the hypothesis may depend on both $p$ and $\mathcal U$. The intended class of spaces and the missing ultrafilter-cover definitions must be recovered before this schematic formulation becomes a fully specified mathematical statement.

Short English statement

Must a space having an $\mathcal O_p$-subcover of every $\omega$-cover, separately for every intended ultrafilter $p$, have a $\gamma$-subcover of every $\omega$-cover? The space class, ultrafilter domain, and precise meaning of $\mathcal O_p$ are unspecified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800655$, source code catalogue entry, title "Selection Principles Milestone Problems --- Problem 28 ([GTpseq, Problem 3.14(2)])." The registry lists Tsaban as proposer, gives the year $2003$, and labels the problem open; this status is supplied metadata, not an independently verified literature assessment.
• Supplied source: Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 28, https://arxiv.org/abs/math/0312182. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source consultation.
• Supplied attribution/context: [GTpseq, Problem 3.14(2)]. The record does not supply the bibliographic expansion of GTpseq.