Selection Principles Milestone Problems — Conjecture 35 ([strongdiags, Conjecture 1])

Definitions and mathematical statement

The supplied record does not specify the class of spaces under consideration or its conventions for admissible covers. The following gives a conditional formulation using conventional cover definitions; these conventions require confirmation against the source.

Let $\mathscr K$ be the intended class of topological spaces. For each $X\in\mathscr K$, let $\mathcal C(X)$ be the intended class of admissible open covers of $X$, with $X\notin\mathcal U$ for every $\mathcal U\in\mathcal C(X)$. Any countability restriction on these covers remains unspecified. Put ${\mathbb{N}}=\{1,2,\ldots\}$. For $n\in{\mathbb{N}}$, define \[ \mathcal O_n(X)=\left\{\mathcal U\in\mathcal C(X):\text{every }F\subseteq X\text{ with }|F|\le n\text{ is contained in some }U\in\mathcal U\right\}, \] and define the family of $\omega$-covers by \[ \Omega(X)=\left\{\mathcal U\in\mathcal C(X):\text{every finite }F\subseteq X\text{ is contained in some }U\in\mathcal U\right\}. \] A cover $\mathcal U$ is large if every point of $X$ belongs to infinitely many distinct members of $\mathcal U$. Define $\mathrm T(X)$ to consist of the large covers $\mathcal U\in\mathcal C(X)$ such that, for all $x,y\in X$, at least one of \[ \{U\in\mathcal U:x\in U,\ y\notin U\},\qquad \{U\in\mathcal U:y\in U,\ x\notin U\} \] is finite. These are the $\tau$-covers under the stated convention.

For families $\mathcal A_n$ of covers and a family $\mathcal B$ of covers, the property $\mathsf S_1(\{\mathcal A_n\}_{n\in{\mathbb{N}}},\mathcal B)$ means that for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ with $\mathcal U_n\in\mathcal A_n$, there exist choices $U_n\in\mathcal U_n$ such that $\{U_n:n\in{\mathbb{N}}\}\in\mathcal B$. Write $\mathsf S_1(\mathcal A,\mathcal B)$ when $\mathcal A_n=\mathcal A$ for every $n$.

Set \[ P(X):=\mathsf S_1(\{\mathcal O_n(X)\}_{n\in{\mathbb{N}}},\mathrm T(X)),\qquad Q(X):=\mathsf S_1(\Omega(X),\mathrm T(X)). \] The supplied Conjecture 35 asserts that $P$ is strictly stronger than $Q$, meaning \[ \left(\forall X\in\mathscr K,\ P(X)\Longrightarrow Q(X)\right) \quad\text{and}\quad \left(\exists X\in\mathscr K,\ Q(X)\land\neg P(X)\right). \] The intended class $\mathscr K$ and admissible-cover conventions must be recovered before this conditional formulation can be treated as an exact source statement.

Short English statement

The conjecture says that choosing one member from each successive $n$-cover to obtain a $\tau$-cover is strictly stronger than making such choices from a sequence of $\omega$-covers. Strictness requires a space satisfying the second property but failing the first. The intended spaces and cover conventions are unspecified in the supplied record.

Sources

• Local UnsolvedMath registry: permanent ID $800662$, exact source problem code catalogue entry. The supplied record labels the problem open; this is a supplied status, not an independently verified mathematical or literature assessment.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Conjecture 35. Supplied source URL: https://arxiv.org/abs/math/0312182. The record lists Tsaban as proposer and 2003 as the proposed year.
• The source-specific attribution/context is [strongdiags, Conjecture 1]. The supplied record does not resolve this bibliography label into a full reference or identify its authors.
• Supplied provenance: access date 2026-10-16; extraction method arxiv_latex. These are supplied provenance statements; no independent source consultation is claimed.