Selection Principles in Mathematics — Problem 39 ([Sakai06])

Definitions and mathematical statement

For a set $X\subseteq{\mathbb{R}}$, give $X$ its subspace topology. Under the standard real-valued interpretation, let \[ C_p(X)=\{f:X\to{\mathbb{R}}:f\text{ is continuous}\} \] with the topology of pointwise convergence, inherited from ${\mathbb{R}}^X$.

For a topological space $Y$ and $y\in Y$, use the conventions \[ \Omega_y=\{A\subseteq Y:y\in\overline A\setminus A\},\qquad O(\Omega_y)=\{U\in\Omega_y:U\text{ is open in }Y\}, \] and \[ \Gamma_y=\{B\subseteq Y:B\text{ is infinite},\ y\notin B,\ B\setminus V\text{ is finite for every neighborhood }V\text{ of }y\}. \] The property $\binom{O(\Omega_y)}{\Gamma_y}$ means that every $U\in O(\Omega_y)$ contains some $B\in\Gamma_y$. A space is $\kappa$-Fréchet when this property holds at every point $y\in Y$; here $\kappa$ is part of the property name, not a cardinal parameter to be determined.

Determine the least cardinality \[ \min\left\{|X|:X\subseteq{\mathbb{R}}\text{ and }C_p(X)\text{ is not }\kappa\text{-Fr\'echet}\right\}. \] Thus the requested failure means that there exist $f\in C_p(X)$ and an open set $U\subseteq C_p(X)$ with $f\in\overline U\setminus U$, such that no infinite subset $B\subseteq U$ has all but finitely many of its elements in every neighborhood of $f$.

The supplied excerpt defines $O(\Omega_y)$ but does not expand $C_p(X)$, $\Omega_y$, $\Gamma_y$, or the binomial selection notation. The formulation above explicitly adopts their standard interpretations; these conventions require confirmation against the cited source.

Short English statement

Find the smallest size of a set of reals $X$ for which the space of continuous real-valued functions on $X$, with pointwise convergence, fails to be $\kappa$-Fréchet. Under the stated conventions, failure occurs when an open set accumulates at a function but contains no infinite subset converging to that function.

Sources

• Local UnsolvedMath registry: permanent ID $800611$, source problem code catalogue entry, AMR Open Problem Lists. The supplied record labels the problem open; this status is not independently verified here.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 39. Supplied source URL: https://arxiv.org/abs/math/0609601. The record attributes the proposal to Tsaban and records the proposed year as 2006.
• Supplied attribution/context: [Sakai06]. The record does not provide the bibliographic entry corresponding to this label.
• Supplied provenance: accessed 2026-10-16; extraction method arxiv_latex. This access date is supplied provenance, not a claim that the source was visited or verified for this draft.