Selection Principles in Mathematics — Problem 17 ([AddQuad])

Definitions and mathematical statement

The supplied excerpt asks whether $\mathsf{S}_{\mathrm{fin}}(\mathrm{B}_\Omega,\mathrm{B}_\Omega)$ is consistently additive. It does not define the ambient class of spaces, the cover conventions, or "additive." The following gives a provisional formulation under the usual interpretation involving sets of reals and closure under finite unions; these conventions require confirmation against the complete source.

Let $X\subseteq{\mathbb{R}}$ have its subspace topology. For this formulation, a cover of $X$ is a countable family $\mathcal U$ of subsets of $X$ such that $\bigcup\mathcal U=X$ and $X\notin\mathcal U$. An $\omega$-cover is a cover such that \[ \forall F\subseteq X\;\left(F\text{ finite}\Longrightarrow\exists U\in\mathcal U\;(F\subseteq U)\right). \] A $\gamma$-cover is an infinite cover $\mathcal U$ such that every $x\in X$ belongs to all but finitely many members of $\mathcal U$. Write $\Omega(X)$ and $\Gamma(X)$ for the families of open $\omega$-covers and open $\gamma$-covers of $X$, respectively. Write $\mathrm{B}_\Omega(X)$ for the family of $\omega$-covers whose members are Borel subsets of $X$.

For cover families $\mathcal A(X)$ and $\mathcal B(X)$, 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 subfamilies $\mathcal F_n\subseteq\mathcal U_n$ such that \[ \bigcup_{n\in{\mathbb{N}}}\mathcal F_n\in\mathcal B(X). \] Here finite subfamilies may be empty. The property $\mathsf{S}_1(\mathcal A,\mathcal B)$ means that for every such sequence there are choices $U_n\in\mathcal U_n$ satisfying $\{U_n:n\in{\mathbb{N}}\}\in\mathcal B(X)$.

Let $P(X)$ mean that $X$ satisfies $\mathsf{S}_{\mathrm{fin}}(\mathrm{B}_\Omega,\mathrm{B}_\Omega)$. Under the provisional interpretation above, the question is whether it is consistent with $\mathsf{ZFC}$ that \[ \forall X,Y\subseteq{\mathbb{R}}\qquad \left(P(X)\land P(Y)\right)\Longrightarrow P(X\cup Y). \] Thus "additive" is interpreted here as preservation under the union of two subsets of the same real line, equivalently under nonempty finite unions.

The excerpt also states: "The problem is also open for $\mathsf{S}_1(\Gamma,\Omega)$ and $\mathsf{S}_{\mathrm{fin}}(\Gamma,\Omega)$." Its antecedent is absent from the supplied record. If that sentence refers to consistent additivity, the corresponding questions replace $P$ in the displayed assertion by each of these two properties. This interpretation of the additional sentence remains unresolved; it is not asserted as an established reading of the source.

Short English statement

Under a provisional sets-of-reals interpretation, ask whether it is consistent with $\mathsf{ZFC}$ that the union of any two sets satisfying $\mathsf{S}_{\mathrm{fin}}(\mathrm{B}_\Omega,\mathrm{B}_\Omega)$ again satisfies that property. The domain and meaning of additivity need source confirmation. An additional sentence mentions $\mathsf{S}_1(\Gamma,\Omega)$ and $\mathsf{S}_{\mathrm{fin}}(\Gamma,\Omega)$, but its missing antecedent prevents a definitive formulation of those additional questions.

Sources

• Local UnsolvedMath registry: permanent ID $800591$, source problem code catalogue entry; title: Selection Principles in Mathematics --- Problem 17 ([AddQuad]). The registry attributes the problem to Tsaban, with proposed year 2006, and supplies the status label "open." This label is supplied metadata, not an independently verified mathematical or literature assessment.
• Supplied source reference: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 17, 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 or verification for this draft.
• Supplied attribution/context marker: [AddQuad]. The record gives no bibliographic expansion of this marker, so its authorship and precise reference remain unresolved.