Selection Principles Milestone Problems — Problem 30 ([GTpseq, Problem 0.6 (reformulated)])

Definitions and mathematical statement

For interpreting limits of sequences indexed by ${\mathbb{N}}$, let $M$ be a collection of ultrafilters on ${\mathbb{N}}$, where an ultrafilter is a maximal proper filter. For a sequence $(a_n)_{n\in{\mathbb{N}}}$ in a topological space $Y$, write \[ p\text{-}\lim_{n\in{\mathbb{N}}}a_n=y \quad\Longleftrightarrow\quad \{n\in{\mathbb{N}}:a_n\in U\}\in p \text{ for every neighborhood }U\text{ of }y. \] The supplied record does not specify whether the ultrafilters must be free or whether $M$ must be nonempty.

Following the definitions attributed in the record to Kombarov, $Y$ is weakly $M$-sequential if \[ \forall A\subseteq Y,\quad A\ne\overline A \ \Longrightarrow\ \exists y\in Y\setminus A\; \exists(a_n)_{n\in{\mathbb{N}}}\in A^{{\mathbb{N}}}\; \exists p\in M:\quad p\text{-}\lim_{n\in{\mathbb{N}}}a_n=y. \] It is strongly $M$-sequential if instead \[ \forall A\subseteq Y,\quad A\ne\overline A \ \Longrightarrow\ \exists y\in Y\setminus A\; \exists(a_n)_{n\in{\mathbb{N}}}\in A^{{\mathbb{N}}}\; \forall p\in M:\quad p\text{-}\lim_{n\in{\mathbb{N}}}a_n=y. \] Thus, in the strong version, the same point and sequence must work for every $p\in M$.

Under the usual real-valued convention, $C_p(X)$ denotes the set of continuous functions $X\to{\mathbb{R}}$, with the subspace topology inherited from the product ${\mathbb{R}}^X$. The supplied record does not explicitly specify this convention or the class of spaces $X$.

Question (source formulation). Assuming that $X$ satisfies the $\delta_M$ property, must $C_p(X)$ be weakly $M$-sequential? Equivalently, under the preceding convention for $C_p(X)$, does the assumption imply that every nonclosed $A\subseteq C_p(X)$ admits $f\in C_p(X)\setminus A$, a sequence $(f_n)_{n\in{\mathbb{N}}}$ in $A$, and $p\in M$ such that \[ \forall x\in X\;\forall\varepsilon>0,\qquad \{n\in{\mathbb{N}}:|f_n(x)-f(x)|<\varepsilon\}\in p? \] Unresolved qualification. The supplied record does not define the $\delta_M$ property or specify its admissible spaces and ultrafilter collections. Consequently, the hypothesis above remains an undefined source predicate, and the question cannot be made fully self-contained from the supplied evidence.

Short English statement

The question asks whether the source's $\delta_M$ property of $X$ forces every nonclosed subset of $C_p(X)$ to contain a sequence with a limit outside that subset along some ultrafilter in $M$. The definition of $\delta_M$, the permitted spaces, and the restrictions on $M$ are missing. The real-valued interpretation of $C_p(X)$ requires confirmation.

Sources

• Local UnsolvedMath registry: permanent ID $800657$, exact source code catalogue entry, supplied complete source record; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry attributes the problem to Tsaban, gives the year 2003, and labels it open. This is a supplied status label, not an independently verified mathematical or literature assessment.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 30. Supplied source URL: https://arxiv.org/abs/math/0312182. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access.
• Supplied attribution/context: [GTpseq, Problem 0.6 (reformulated)]. The full bibliographic reference and the original formulation are not supplied.
• The weak and strong $M$-sequentiality definitions are attributed to Kombarov [Km]. The full bibliographic reference for [Km] is not supplied.