Selection Principles in Mathematics — Problem 24 ([AddQuad])

Definitions and mathematical statement

The supplied question concerns the five properties \[ \mathsf{S}_1(\mathrm{T},\mathrm{T}),\quad \mathsf{S}_{\mathrm{fin}}(\mathrm{T},\mathrm{T}),\quad \mathsf{S}_1(\Gamma,\mathrm{T}),\quad \mathsf{S}_{\mathrm{fin}}(\Gamma,\mathrm{T}),\quad \mathsf{U}_{\mathrm{fin}}(\mathcal{O},\mathrm{T}), \] and their Borel versions. The excerpt does not specify the class of spaces, cover conventions, or meaning of additivity. The following gives a conditional formulation using conventional countable-cover definitions; these conventions require confirmation against the source.

Let $X$ be a topological space and let ${\mathbb{N}}=\{1,2,\ldots\}$. A countable cover of $X$ is a finite or countably infinite family $\mathcal U$ of subsets of $X$ such that $\bigcup\mathcal U=X$ and $X\notin\mathcal U$. Let $\mathcal O(X)$ denote the family of countable open covers under this convention. A cover is large if each point belongs to infinitely many of its members. Define $\Gamma(X)$ to consist of the infinite members $\mathcal U$ of $\mathcal O(X)$ such that \[ \forall x\in X,\qquad \left|\{U\in\mathcal U:x\notin U\}\right|<\infty. \] Define $\mathrm T(X)$ to consist of the large members $\mathcal U$ of $\mathcal O(X)$ such that, for every $x,y\in X$, \[ \left|\{U\in\mathcal U:x\in U,\ y\notin U\}\right|<\infty \quad\text{or}\quad \left|\{U\in\mathcal U:y\in U,\ x\notin U\}\right|<\infty. \] These are the open $\gamma$-covers and open $\tau$-covers, respectively.

For families $\mathscr A(X)$ and $\mathscr B(X)$ of covers, $\mathsf S_1(\mathscr A,\mathscr B)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ from $\mathscr A(X)$, there exist $U_n\in\mathcal U_n$ such that \[ \{U_n:n\in{\mathbb{N}}\}\in\mathscr B(X). \] The property $\mathsf S_{\mathrm{fin}}(\mathscr A,\mathscr B)$ means that for every such sequence there exist finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \bigcup_{n\in{\mathbb{N}}}\mathcal F_n\in\mathscr B(X). \] The property $\mathsf U_{\mathrm{fin}}(\mathscr A,\mathscr B)$ means that for every such sequence in which no $\mathcal U_n$ has a finite subcover, there exist finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \left\{\bigcup\mathcal F_n:n\in{\mathbb{N}}\right\}\in\mathscr B(X). \] Here selected families are treated as sets, so repeated selected subsets do not count as distinct cover members.

For the corresponding Borel formulation, let $\mathcal B(X)$ be the countable covers by Borel subsets of $X$, again excluding $X$ as a member. Let $\mathcal B_\Gamma(X)$ and $\mathcal B_{\mathrm T}(X)$ satisfy the same respective pointwise conditions as $\Gamma(X)$ and $\mathrm T(X)$. Under this interpretation, the five Borel versions are \[ \begin{gathered} \mathsf S_1(\mathcal B_{\mathrm T},\mathcal B_{\mathrm T}),\qquad \mathsf S_{\mathrm{fin}}(\mathcal B_{\mathrm T},\mathcal B_{\mathrm T}),\\ \mathsf S_1(\mathcal B_\Gamma,\mathcal B_{\mathrm T}),\qquad \mathsf S_{\mathrm{fin}}(\mathcal B_\Gamma,\mathcal B_{\mathrm T}),\qquad \mathsf U_{\mathrm{fin}}(\mathcal B,\mathcal B_{\mathrm T}). \end{gathered} \]

Question, with the unresolved scope made explicit. Are any of the five listed open-cover properties, or any of their Borel versions, additive? If additivity means preservation under binary unions of subspaces, fix an ambient topological space $Z$ and an intended class $\mathscr D$ of its subspaces that is closed under finite unions. For each listed property $P$, the assertion to be decided is \[ \forall X,Y\in\mathscr D,\qquad \left(P(X)\land P(Y)\right)\Longrightarrow P(X\cup Y), \] where each space carries its subspace topology. The source excerpt does not determine $Z$, $\mathscr D$, or whether this binary-union interpretation is intended. In particular, preservation under countable unions is a separate possible target and is not asserted to be equivalent. Thus the displayed framework is conditional, and a uniquely specified formulation requires the missing source conventions.

Short English statement

The supplied Problem 24 asks which, if any, of five selection principles involving $\tau$-covers, or their Borel versions, are additive. Under a binary-union interpretation, this asks whether the union of two spaces with a given property must also have that property. The intended class of spaces and the precise cover and additivity conventions remain unspecified in the supplied excerpt.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800596$, exact source problem code catalogue entry; upstream ID $800596$, revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry attributes the problem to Tsaban, with proposed year $2006$, and labels its status as open. This is a supplied status label, not an independently verified assessment.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 24. Supplied source URL: https://arxiv.org/abs/math/0609601. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of access during this preparation.
• The supplied title and background associate the problem with the context label [AddQuad]. No bibliographic expansion or explanation of that label is supplied.