Selection Principles Milestone Problems — Problem 44 ([SakaiRez, Question 3.6])

Definitions and mathematical statement

For a topological space $X$, use the conventional real-valued notation \[ C_p(X)=\{f:X\to{\mathbb{R}}:f\text{ is continuous}\}, \] with the topology of pointwise convergence. A basic neighborhood of $f\in C_p(X)$ is \[ [f;K,\varepsilon]=\{g\in C_p(X):|g(k)-f(k)|<\varepsilon\text{ for every }k\in K\}, \] where $K\subseteq X$ is finite and $\varepsilon>0$.

A topological space $Y$ has the Reznicenko property if, for every $A\subseteq Y$ and every $y\in\overline A\setminus A$, there exist pairwise disjoint nonempty finite sets $F_n\subseteq A$, indexed by $n=1,2,\ldots$, such that every neighborhood of $y$ meets all but finitely many of the sets $F_n$.

The supplied Problem 44 asks: \[ \text{Does }C_p(\mathbb{N})\text{ have the Reznicenko property?} \] The supplied extract does not define the source notation $\mathbb{N}$ or its topology. If it means the natural numbers with the discrete topology, then $C_p(\mathbb{N})={\mathbb{R}}^{\mathbb{N}}$ with the product topology, and the question becomes whether the following assertion holds: for every $A\subseteq{\mathbb{R}}^{\mathbb{N}}$ and every $f\in\overline A\setminus A$, there are pairwise disjoint nonempty finite sets $F_n\subseteq A$ such that \[ \begin{split} &\text{for every finite }K\subseteq\mathbb{N}\text{ and every }\varepsilon>0, \text{ there exists }m\geq1\text{ such that}\\ &\qquad\text{for every }n\geq m\text{ there exists }g\in F_n \text{ with }|g(k)-f(k)|<\varepsilon\text{ for all }k\in K. \end{split} \] This specialization is conditional on the discrete-natural-number interpretation. The intended domain must be checked against the source's notation before this draft can be treated as an unambiguous transcription. No answer to the question is asserted.

Short English statement

The source asks whether $C_p(\mathbb{N})$ has the Reznicenko property: at each point in the closure of a set that does not contain the point, one can find disjoint nonempty finite subsets that eventually meet every neighborhood. The meaning and topology of the source's $\mathbb{N}$ require verification; the discrete-natural-number formulation above is conditional.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $800671$, source problem code catalogue entry, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 44. 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 statements, not an independent source visit.
• The supplied attribution/context is explicitly [SakaiRez, Question 3.6]. No full bibliographic identification of SakaiRez is supplied. The registry lists Tsaban as proposer and 2003 as the proposed year.
• The registry labels this record open, with update date 2026-10-16. This is supplied status metadata, not an independently verified literature assessment.