Selection Principles Milestone Problems — Problem 38 ([SchLuzin, Problem 4])

Definitions and mathematical statement

For a topological space $X$, write $\overline{U}^{\,X}$ for the closure of $U$ in $X$. Define \[ \mathcal K(X)=\left\{\mathcal U\subseteq\{U\subseteq X:U\text{ is open in }X\}:\ X=\bigcup_{U\in\mathcal U}\overline{U}^{\,X}\right\}. \] Define $\mathcal K_\Omega(X)$ to consist of those $\mathcal U\in\mathcal K(X)$ such that \[ (\forall U\in\mathcal U)\ \overline{U}^{\,X}\ne X, \qquad (\forall F\subseteq X\text{ finite})(\exists U\in\mathcal U)\ F\subseteq\overline{U}^{\,X}. \] Thus no member of $\mathcal U$ is dense in $X$. No countability restriction on these families is stated in the supplied record.

For a family $\mathcal U$, let $[\mathcal U]^2$ denote its set of two-element subsets. The partition relation \[ \mathcal K_\Omega(X)\longrightarrow\left(\mathcal K(X)\right)^2_2 \] means that for every $\mathcal U\in\mathcal K_\Omega(X)$ and every coloring $c:[\mathcal U]^2\to\{0,1\}$, there exist $\mathcal V\subseteq\mathcal U$ and $i\in\{0,1\}$ such that $\mathcal V\in\mathcal K(X)$ and \[ (\forall p\in[\mathcal V]^2)\ c(p)=i. \]

The supplied source reports that this partition relation implies that $X$ is a Luzin set, attributing that implication to $[\mathrm{SchLuzin}]$. Its question, attributed to $[\mathrm{SchLuzin},\text{ Problem }4]$, is whether the converse holds, and hence whether \[ X\text{ is a Luzin set} \quad\Longleftrightarrow\quad \mathcal K_\Omega(X)\longrightarrow\left(\mathcal K(X)\right)^2_2. \] This proposed equivalence must be read within the source's intended domain of spaces; that domain and the definition of "Luzin set" are not supplied. Under the usual real-line convention, a Luzin set is an uncountable set $X\subseteq{\mathbb{R}}$ such that $X\cap M$ is at most countable for every meager subset $M$ of ${\mathbb{R}}$, where meager means a countable union of nowhere dense sets. Under that convention, $X$ would carry the subspace topology from ${\mathbb{R}}$. This convention is recorded as a possible interpretation, not as an established hypothesis of the supplied source. The missing domain and convention must be resolved before the displayed equivalence is a fully specified assertion.

Short English statement

The question asks whether every Luzin set has the property that every two-coloring of pairs from a family in $\mathcal K_\Omega(X)$ admits a monochromatic subfamily whose closures cover $X$. The source reports the reverse implication. Its intended class of spaces and definition of Luzin set require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $800665$, source problem code catalogue entry; supplied title: "Selection Principles Milestone Problems --- Problem 38 ([SchLuzin, Problem 4])." The registry attributes the record to Tsaban, with proposed year 2003, and labels it open; this supplied status is not an independently verified current assessment.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 38. Supplied source URL: https://arxiv.org/abs/math/0312182. The record gives access date 2026-10-16 and extraction method arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The source attributes the reported implication to $[\mathrm{SchLuzin}]$ and the question to $[\mathrm{SchLuzin},\text{ Problem }4]$. No full bibliographic entry or separate URL for this reference is supplied.