Selection Principles in Mathematics — Problem 27

Definitions and mathematical statement

The supplied excerpt does not specify the underlying class of spaces or its cover conventions. The following gives a conditional formulation using countable open covers; these conventions and the intended class of spaces require verification against the source.

Fix an infinite topological space $X$. A cover here is an at most countable family $\mathcal U$ of open subsets of $X$ such that $\bigcup\mathcal U=X$ and $X\notin\mathcal U$. Write $\mathcal O(X)$ for these covers. Define the following subclasses:

• $\Omega(X)$ consists of covers $\mathcal U$ such that every finite subset of $X$ is contained in some member of $\mathcal U$.
• $\Gamma(X)$ consists of infinite covers $\mathcal U$ such that $\{U\in\mathcal U:x\notin U\}$ is finite for each $x\in X$.
• $\mathrm T(X)$ consists of covers $\mathcal U$ such that each point belongs to infinitely many members and, for every $x,y\in X$, at least one of the sets \[ \{U\in\mathcal U:x\in U,\ y\notin U\},\qquad \{U\in\mathcal U:y\in U,\ x\notin U\} \] is finite. These are called $\tau$-covers.

A cover $\mathcal U$ is multifinite if it admits a partition into infinitely many finite subfamilies, each of which covers $X$. For a class $\mathcal B(X)$ of covers, let $\gimel(\mathcal B)(X)$ consist of those covers $\mathcal U$ that are either multifinite or admit a partition $\mathcal P$ into nonempty finite subfamilies satisfying \[ \left\{\bigcup\mathcal F:\mathcal F\in\mathcal P\right\}\setminus\{X\}\in\mathcal B(X). \] These definitions of multifiniteness and $\gimel$ are an explicit interpretation of notation left undefined in the supplied excerpt and require source confirmation.

For classes $\mathcal A(X),\mathcal B(X)$ of covers, the property $\mathsf S_{\mathrm{fin}}(\mathcal A,\mathcal B)$ means that for every sequence $(\mathcal U_n)_{n\in\mathbb N}$ from $\mathcal A(X)$, there are finite, possibly empty, subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \bigcup_{n\in\mathbb N}\mathcal F_n\in\mathcal B(X). \] The property $\mathsf U_{\mathrm{fin}}(\mathcal A,\mathcal B)$ means that for every such sequence in which no $\mathcal U_n$ has a finite subcover, there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal F_n:n\in\mathbb N\right\}\in\mathcal B(X). \]

Problem 27, attributed to Tsaban (2006). Determine whether, for every $X$ in the source's intended class of spaces, \[ X\models\mathsf U_{\mathrm{fin}}(\mathcal O,\mathrm T) \quad\Longleftrightarrow\quad X\models\mathsf S_{\mathrm{fin}}(\Omega,\gimel(\mathrm T)). \] The intended class of spaces remains unspecified in the supplied record; the displayed question must not be read as a verified assertion that the source quantifies over all topological spaces.

For context, the supplied source states \[ \gimel(\Gamma)\subseteq\gimel(\mathrm T)\subseteq\gimel(\Omega), \] and reports the following equalities of selection properties: \begin{align*} \mathsf S_{\mathrm{fin}}(\mathcal O,\mathcal O) &=\mathsf S_{\mathrm{fin}}(\Omega,\mathcal O) &&\text{[coc1]},\\ \mathsf U_{\mathrm{fin}}(\mathcal O,\Gamma) &=\mathsf S_{\mathrm{fin}}(\Omega,\gimel(\Gamma)) &&\text{[coc7]},\\ \mathsf U_{\mathrm{fin}}(\mathcal O,\Omega) &=\mathsf S_{\mathrm{fin}}(\Omega,\gimel(\Omega)) &&\text{[coc8]}. \end{align*} It says these results are generalized in [GlCovs] and that a positive answer to the problem is consistent, citing [SF2]. The excerpt supplies neither the consistency hypotheses nor the bibliographic entries for these labels. These are preserved source reports, not independently verified results or a specified consistency theorem.

Short English statement

Tsaban's Problem 27 asks whether selecting finite subfamilies from open covers without finite subcovers to obtain a $\tau$-cover by their unions is equivalent to selecting finite subfamilies from arbitrary $\omega$-covers whose combined members belong to $\gimel(\mathrm T)$. The supplied registry labels the problem open and reports that a positive answer is consistent. The intended spaces, cover conventions, and consistency hypotheses need source verification.

Sources

• Local UnsolvedMath registry, permanent ID $800599$, source problem code catalogue entry, title Selection Principles in Mathematics --- Problem 27. The supplied record attributes the problem to Tsaban, with proposed year 2006, and labels it open. Its supplied creation and update timestamps are 2026-10-16T14:00:00Z; this status has not been independently verified.
• Supplied source attribution: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 27, https://arxiv.org/abs/math/0609601. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of access during this drafting.
• Reference labels retained exactly from the supplied excerpt: [coc1], [coc7], [coc8], [GlCovs], and [SF2]. Full bibliographic records are not supplied. In particular, the consistency report is attributed there to [SF2].
• Supplied upstream identity: $800599$; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.