Selection Principles in Mathematics — Problem 20 (Miller, Tsaban)

Definitions and mathematical statement

For a topological space $Y$, let $\Omega(Y)$ denote the family of open $\omega$-covers of $Y$: a family $\mathcal U$ of open subsets of $Y$ belongs to $\Omega(Y)$ if $Y\notin\mathcal U$ and every finite subset of $Y$ is contained in some member of $\mathcal U$. Let $\Gamma(Y)$ denote the family of infinite open $\gamma$-covers of $Y$: using the nontrivial-cover convention $Y\notin\mathcal U$, these are the infinite families $\mathcal U$ of open subsets of $Y$ such that every point of $Y$ belongs to all but finitely many members of $\mathcal U$. The property $\mathsf S_1(\Omega,\Gamma)$ means that for every sequence $(\mathcal U_n)_{n\geq 1}$ with $\mathcal U_n\in\Omega(Y)$, there are choices $U_n\in\mathcal U_n$ such that \[ \{U_n:n\geq 1\}\in\Gamma(Y). \] In particular, the selected family must be infinite.

For infinite subsets $a,b\subseteq{\mathbb{N}}$, write $a\subseteq^* b$ if $a\setminus b$ is finite. A family $\mathcal A\subseteq[{\mathbb{N}}]^{\aleph_0}$ has the strong finite intersection property if every finite intersection of members of $\mathcal A$ is infinite. An infinite pseudointersection of $\mathcal A$ is an infinite set $a\subseteq{\mathbb{N}}$ such that $a\subseteq^* b$ for every $b\in\mathcal A$. The pseudointersection number is \[ \mathfrak p=\min\left\{|\mathcal A|:\mathcal A\subseteq[{\mathbb{N}}]^{\aleph_0},\ \mathcal A\text{ has the strong finite intersection property and has no infinite pseudointersection}\right\}. \]

The supplied record does not explicitly specify the ambient space or the admissible class of $X$ and $D$. Its source title suggests sets of reals, but this does not determine the missing hypotheses. To make the union meaningful, let $X,D\subseteq T$ in the intended common ambient topological space $T$, and give $X$ and $X\cup D$ their subspace topologies. The following questions are conditional on confirming that intended ambient setting and the cover conventions above; they are not assertions quantified over all topological spaces.

Question 1 (Miller, Tsaban). For all admissible $X,D\subseteq T$, is the implication \[ X\models\mathsf S_1(\Omega,\Gamma)\quad\text{and}\quad |D|<\mathfrak p \quad\Longrightarrow\quad X\cup D\models\mathsf S_1(\Omega,\Gamma) \] true?

Question 2. Assuming Martin's Axiom, is it true for all admissible $X,D\subseteq T$ that \[ X\models\mathsf S_1(\Omega,\Gamma)\quad\text{and}\quad |D|=\aleph_1 \quad\Longrightarrow\quad X\cup D\models\mathsf S_1(\Omega,\Gamma)? \] The second question is read as replacing the cardinality restriction in the first question while retaining the assumption on $X$. The source does not state an additional hypothesis concerning the continuum or specify whether its use of Martin's Axiom includes a separate assumption negating the continuum hypothesis.

Short English statement

The problem asks whether adjoining fewer than $\mathfrak p$ points preserves $\mathsf S_1(\Omega,\Gamma)$, and whether adjoining $\aleph_1$ points preserves it under Martin's Axiom. The intended ambient class of spaces and the precise axiom conventions require confirmation. The supplied registry labels the problem open; that label is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800593$, source problem code catalogue entry, in the AMR Open Problem Lists collection. The record names Tsaban as proposer and gives the year 2006; its title and background attribute the problem to Miller and Tsaban. The supplied status is open.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 20. Supplied source URL: https://arxiv.org/abs/math/0609601. The record reports extraction by arxiv_latex and access on 2026-10-16. These are supplied provenance statements, not a claim of independent source access or verification.