Selection Principles Milestone Problems — Problem 12 (\Cite{\mathbb{S} 4}{spm1})

Definitions and mathematical statement

The supplied record does not specify the ambient class of spaces, the permitted cardinalities of covers, or the definitions of its cover classes. The following gives the usual open-cover interpretation, with these unresolved qualifications made explicit. Let $\mathscr X$ denote the intended, unspecified class of topological spaces. For $X\in\mathscr X$, let $\mathcal O_*(X)$ denote the source's intended class of admissible open covers. Under the usual convention, each $\mathcal U\in\mathcal O_*(X)$ is a family of open subsets of $X$ such that $\bigcup\mathcal U=X$ and $X\notin\mathcal U$. Whether admissible covers must be countable is unresolved in the supplied record.

Using this interpretation, define the following subclasses of $\mathcal O_*(X)$:

• $\Omega(X)$ consists of the $\omega$-covers: for every finite subset $F\subseteq X$, there is $U\in\mathcal U$ with $F\subseteq U$.
• $\Gamma(X)$ consists of the $\gamma$-covers: $\mathcal U$ is infinite and, for every $x\in X$, the family $\{U\in\mathcal U:x\notin U\}$ is finite.
• $\mathrm T(X)$ consists of the $\tau$-covers: for every $x\in X$, the family $\{U\in\mathcal U:x\in U\}$ is infinite, 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.

For classes $\mathcal A(X)$ and $\mathcal B(X)$ of covers, write \[ X\models\binom{\mathcal A}{\mathcal B} \quad\Longleftrightarrow\quad (\forall\mathcal U\in\mathcal A(X)) (\exists\mathcal V\subseteq\mathcal U)\; \mathcal V\in\mathcal B(X). \]

The source question is whether, throughout its intended class of spaces and with its intended cover conventions, \[ (\forall X\in\mathscr X)\qquad X\models\binom{\Omega}{\Gamma} \quad\Longleftrightarrow\quad X\models\binom{\Omega}{\mathrm T}. \] The inclusion $\Gamma(X)\subseteq\mathrm T(X)$ gives the forward implication. Thus the requested implication is that every space in the intended class for which every admissible $\omega$-cover contains a $\tau$-subcover also has a $\gamma$-subcover of every admissible $\omega$-cover. The unspecified ambient class and admissibility conventions must be recovered before this becomes a fully determined assertion.

For context, $\mathsf S_1(\Omega,\Gamma)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ of members of $\Omega(X)$, one can choose $U_n\in\mathcal U_n$ for each $n$ so that $\{U_n:n\in{\mathbb{N}}\}\in\Gamma(X)$. The supplied source attributes to reference [GN], in its intended setting, the equivalence \[ \mathsf S_1(\Omega,\Gamma)\Longleftrightarrow\binom{\Omega}{\Gamma}. \] It also reports that all examples then considered which failed $\binom{\Omega}{\Gamma}$ also failed $\binom{\Omega}{\mathrm T}$. This is historical source context, not a resolution of the question or an independently verified present-day assessment.

Short English statement

In the source's intended class of spaces, does the property that every open $\omega$-cover contains a $\tau$-subcover imply that every such cover contains a $\gamma$-subcover? The converse follows from $\Gamma\subseteq\mathrm T$. The supplied record leaves the ambient spaces and cover cardinality conventions unspecified. Its label open is supplied registry metadata, not an independently verified current status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800640$, exact source problem code catalogue entry, in the AMR Open Problem Lists collection. The record attributes the problem to Tsaban, with proposed year 2003, and labels it open. Its supplied creation and update date is 2026-10-16; upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 12. Supplied source URL: 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 access or verification.
• The title and background retain an attribution/context pointer to Section 4 of the reference labelled spm1. No bibliographic identification of that reference is supplied.
• The problem text cites reference labels [tau], [tautau], [splittability], [strongdiags], [Hdim], and [MilNonGamma] for studies of $\tau$-covers. It attributes $\mathsf S_1(\Omega,\Gamma)\Longleftrightarrow\binom{\Omega}{\Gamma}$ to [GN]. The supplied record does not identify these references bibliographically.