Selection Principles in Mathematics — Problem 57 (Bartoszy\'nski)

Definitions and mathematical statement

The supplied formulation has unresolved ambient-space and section restrictions. The following gives its logical form with the missing ambient data displayed explicitly; it does not identify those data with a verified source formulation.

Let $(S,\tau)$ be a topological space and let $\preceq$ be a quasiorder on $S$. A set $A\subseteq S$ is nowhere dense if $\operatorname{int}_{\tau}(\overline{A}^{\tau})=\varnothing$, and is meager if it is a countable union of nowhere dense sets. Write $\mathcal M(S,\tau)$ for the family of meager subsets of $S$. A set $Y\subseteq S$ is dominating relative to $\preceq$ if \[ \forall f\in S\ \exists y\in Y\quad f\preceq y. \] For $H\subseteq S\times S$ and $x\in S$, define its vertical section by \[ H_x=\{z\in S:(x,z)\in H\}. \] The question has the schematic form: for every $X,Y\subseteq S$ satisfying \[ X\notin\mathcal M(S,\tau) \quad\text{and}\quad \forall f\in S\ \exists y\in Y\ (f\preceq y), \] does there exist a set $H\subseteq S\times S$, Borel in the product topology, such that \[ \forall M\in\mathcal M(S,\tau)\ \exists x\in X\cup Y\quad M\subseteq H_x? \] The supplied record writes $X,Y\subseteq\mathbb N$ and $H\subseteq\mathbb N\times\mathbb N$, but specifies neither a topology nor a domination relation. In particular, it does not explicitly specify the Baire space $\mathbb N^{\mathbb N}$, whose usual eventual-domination relation is \[ f\leq^{*}g \quad\Longleftrightarrow\quad \exists n_0\in\mathbb N\ \forall n\geq n_0\quad f(n)\leq g(n). \] This possible interpretation remains unconfirmed. Moreover, the supplied statement imposes no meagerness or other restriction on the sections $H_x$, and does not exclude the full product as $H$. No additional section restriction has been inserted into the displayed question; the intended substantive target therefore requires clarification.

Short English statement

For permanent UnsolvedMath ID $800628$, the supplied question asks whether a nonmeager set together with a dominating set can index sections of one Borel relation that contain every meager set. The ambient space, topology, domination relation, and any intended restriction on those sections remain unresolved in the supplied formulation.

Sources

• Local registry: UnsolvedMath, permanent ID $800628$, source code catalogue entry, collection AMR Open Problem Lists. The supplied record labels the problem open; this is a supplied registry status, not an independently verified mathematical or literature assessment.
• Supplied source reference: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 57, https://arxiv.org/abs/math/0609601. The registry lists Tsaban as proposer and 2006 as the proposed year; the title and background attribute the problem or its context to Bartoszyński.
• Supplied provenance: source access date 2026-10-16; extraction method arxiv_latex; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. These are supplied provenance claims; the source and its notation have not been independently checked for this draft.