Selection Principles Milestone Problems — Problem 36

Definitions and mathematical statement

The supplied excerpt leaves the underlying class of spaces and the cover conventions unspecified. The following is a parameterized formulation requiring confirmation against the source. Fix a topological space $X$, and put ${\mathbb{N}}=\{1,2,\ldots\}$. Let $\mathscr C(X)$ denote the admissible open covers of $X$. For this formulation, use the nontrivial-cover convention: each member of a cover is open, its union is $X$, and $X$ itself is not a member. Whether admissible covers must also be countable is unresolved; the same choice of convention must be used throughout.

For $n\in{\mathbb{N}}$, define \[ \mathcal O_n(X)=\left\{\mathcal U\in\mathscr C(X):\ \forall F\subseteq X\text{ with }|F|\le n,\ \exists U\in\mathcal U\ (F\subseteq U)\right\}. \] Define the family of admissible $\omega$-covers by \[ \Omega(X)=\left\{\mathcal U\in\mathscr C(X):\ \forall F\subseteq X\text{ finite},\ \exists U\in\mathcal U\ (F\subseteq U)\right\}. \] Under the usual groupability interpretation, $\Omega^{gp}(X)$ consists of those $\mathcal U\in\Omega(X)$ admitting a partition \[ \mathcal U=\bigsqcup_{k\in{\mathbb{N}}}\mathcal B_k \] into nonempty finite families such that \[ \forall F\subseteq X\text{ finite},\ \exists k_0\in{\mathbb{N}}\ \forall k\ge k_0\ \exists U\in\mathcal B_k\quad F\subseteq U. \] This interpretation of the superscript $gp$, like the cover conventions, is not defined in the supplied excerpt and requires source confirmation.

For $\mathfrak V\in\{\Omega,\Omega^{gp}\}$, the property $\mathsf S_{\mathrm{fin}}(\Omega,\mathfrak V)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ from $\Omega(X)$, there are finite, possibly empty, families $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \bigcup_{n\in{\mathbb{N}}}\mathcal F_n\in\mathfrak V(X). \] In the game $\mathsf G_{\mathrm{fin}}(\{\mathcal O_n\}_{n\in{\mathbb{N}}},\mathfrak V)$, inning $n$ consists of ONE choosing $\mathcal U_n\in\mathcal O_n(X)$ and TWO choosing a finite, possibly empty, subfamily $\mathcal F_n\subseteq\mathcal U_n$. TWO wins exactly when $\bigcup_{n\in{\mathbb{N}}}\mathcal F_n\in\mathfrak V(X)$; otherwise ONE wins. A winning strategy for ONE must win against every sequence of legal responses by TWO.

The question attributed to Tsaban, Problem 36 (2003), is whether, for each $\mathfrak V\in\{\Omega,\Omega^{gp}\}$, \[ \text{ONE has no winning strategy in }\mathsf G_{\mathrm{fin}}(\{\mathcal O_n(X)\}_{n\in{\mathbb{N}}},\mathfrak V(X)) \quad\Longleftrightarrow\quad \mathsf S_{\mathrm{fin}}(\Omega(X),\mathfrak V(X)). \] The intended quantification over spaces $X$ is missing from the supplied record. Thus this is a statement template under explicitly stated conventions, not a verified assertion that the source asks this for all topological spaces. The supplied registry labels the problem open; that label is not an independently verified assessment.

Short English statement

For each of the targets $\Omega$ and $\Omega^{gp}$, must the finite selection property for sequences of $\omega$-covers be equivalent to ONE having no winning strategy when ONE must play an open $n$-cover in inning $n$? The intended class of spaces and the precise cover conventions require source confirmation.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $800663$, exact source problem code catalogue entry, title "Selection Principles Milestone Problems --- Problem 36." The record attributes the problem to Tsaban (2003) and labels it open; this status is supplied metadata, not independently verified.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 36. Supplied source URL: https://arxiv.org/abs/math/0312182. The background records extraction by arxiv_latex and an access date of 2026-10-16. That access date is supplied provenance; no independent source visit or literature verification is claimed here.