Selection Principles in Mathematics — Problem 26 ([tautau])

Definitions and mathematical statement

Let $X=\{0,1\}^{{\mathbb{N}}}$, equipped with the product topology in which $\{0,1\}$ is discrete. Write $[{\mathbb{N}}]^{\aleph_0}$ for the set of infinite subsets of ${\mathbb{N}}$. For $a,b\subseteq{\mathbb{N}}$, write $a\subseteq^*b$ if $a\setminus b$ is finite. A family $Y\subseteq[{\mathbb{N}}]^{\aleph_0}$ is linearly refinable if there is a choice of infinite sets $\widehat y\subseteq y$, one for each $y\in Y$, such that \[ \forall y,z\in Y\quad \left(\widehat y\subseteq^*\widehat z\ \lor\ \widehat z\subseteq^*\widehat y\right). \] A countable open cover $\mathcal U=\{U_n:n\in{\mathbb{N}}\}$ of $X$ is large if every $x\in X$ belongs to infinitely many members of $\mathcal U$. Define its incidence sets by \[ a_x=\{n\in{\mathbb{N}}:x\in U_n\}\qquad(x\in X). \] The cover is a $\tau^*$-cover if it is large and $\{a_x:x\in X\}$ is linearly refinable. Let $\mathrm T^*(X)$ denote the collection of countable open $\tau^*$-covers of $X$. An infinite open cover $\mathcal V$ is a $\gamma$-cover if \[ \forall x\in X\quad \{V\in\mathcal V:x\notin V\}\text{ is finite}. \] Let $\Gamma(X)$ denote the collection of countable open $\gamma$-covers. The notation $\binom{\mathrm T^*}{\Gamma}$ means that every member of $\mathrm T^*(X)$ contains a subfamily belonging to $\Gamma(X)$.

Question (source Problem 26). Does the Cantor space $X$ satisfy \[ \forall\mathcal U\in\mathrm T^*(X)\quad \exists\mathcal V\subseteq\mathcal U\quad \mathcal V\in\Gamma(X)? \] Equivalently, for a cover enumerated without repetitions, must there exist an infinite $I\subseteq{\mathbb{N}}$ such that \[ \forall x\in X\quad I\subseteq^*a_x? \] Here the infinite refinements witnessing linear refinability may depend on $x$, whereas the desired set $I$ must work simultaneously for every $x\in X$.

Unresolved conventions and context. The definitions of large cover, $\Gamma$, and the binomial selection notation above use their standard meanings; the supplied record does not spell them out. This formulation treats covers as families of distinct sets and does not impose the additional condition $X\notin\mathcal U$. The intended convention about proper cover members and repeated terms requires confirmation from the source. As background, the record also asserts that every analytic space satisfies $\binom{\mathrm T}{\Gamma}$, with reference label $[\mathrm{tau}]$. It does not define $\mathrm T$, specify the scope of that assertion, or supply the bibliography entry for that label; the assertion is retained as supplied context, not independently verified.

Short English statement

Does every countable open $\tau^*$-cover of the Cantor space contain an infinite subcover in which each point belongs to all but finitely many members? A $\tau^*$-cover allows each point's infinite incidence set to be thinned so that the resulting sets are linearly ordered modulo finite differences. The registry labels this problem open; cover conventions and unresolved reference labels require human review.

Sources

• Local UnsolvedMath registry, permanent ID $800598$, exact source problem code catalogue entry, AMR Open Problem Lists. The record attributes the problem to Tsaban, with proposed year 2006, and identifies it as Problem 26, with attribution/context label [tautau]. Its supplied status is open; the record was updated on 2026-10-16. This is supplied registry metadata, not an independently verified status assessment.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 26: https://arxiv.org/abs/math/0609601. The record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; no independent source visit or verification is claimed.
• The supplied contextual assertion about analytic spaces cites [tau], and the problem title/context uses [tautau]. Corresponding bibliography entries are absent from the supplied record, so their authorship and publication details remain unresolved.