Selection Principles Milestone Problems — Problem 29 ([GT1], [GTpseq, Problem 0.10])

Definitions and mathematical statement

Let $p$ be an ultrafilter on the sequence index set ${\mathbb{N}}$. For a sequence $(y_n)_{n\in{\mathbb{N}}}$ in a topological space $Y$, define \[ y=p\text{-}\lim_{n\in{\mathbb{N}}}y_n \quad\Longleftrightarrow\quad \{n\in{\mathbb{N}}:y_n\in U\}\in p \text{ for every neighborhood }U\text{ of }y. \] Using the usual characterization of sequential spaces by sequentially closed subsets, the supplied definition of $p$-sequentiality reads as follows: $Y$ is $p$-sequential if every subset $A\subseteq Y$ satisfying \[ \forall (a_n)_{n\in{\mathbb{N}}}\in A^{{\mathbb{N}}}\ \forall y\in Y, \qquad y=p\text{-}\lim_{n\in{\mathbb{N}}}a_n\ \Longrightarrow\ y\in A \] is closed in $Y$.

Under the conventional real-valued interpretation, $C_p(X)$ denotes the set $C(X,{\mathbb{R}})$ of continuous functions with the topology inherited from the product ${\mathbb{R}}^X$. The subscript in $C_p(X)$ denotes pointwise convergence, whereas the ultrafilter $p$ is the parameter in $p$-sequentiality. The supplied record does not specify the separation assumptions on $X$ or explicitly confirm this function-space convention.

An open $\omega$-cover of $X$, under the usual convention, is a family $\mathcal U$ of open subsets of $X$ such that $X\notin\mathcal U$ and every finite subset of $X$ is contained in some member of $\mathcal U$. Write $\Omega(X)$ for the intended class of such covers; whether this class is restricted to countable covers is not stated in the record. For classes of covers $\mathscr A$ and $\mathscr B$, the usual interpretation of $\binom{\mathscr A}{\mathscr B}$ is that every member of $\mathscr A$ has a subfamily belonging to $\mathscr B$.

The supplied question asks whether, for the intended spaces $X$ and ultrafilters $p$, \[ C_p(X)\text{ is }p\text{-sequential} \quad\Longrightarrow\quad \forall\mathcal U\in\Omega(X)\ \exists\mathcal V\subseteq\mathcal U, \qquad \mathcal V\in\mathcal O_p(X). \] Here $\mathcal O_p(X)$ retains the source notation: its definition is absent from the supplied record. The record also does not say whether $p$ must be free or whether the implication is intended for every ultrafilter in a particular class. Consequently, this is a statement schema with unresolved source qualifications, rather than a fully specified assertion.

Short English statement

The question asks whether $p$-sequentiality of the pointwise function space $C_p(X)$ forces every open $\omega$-cover of $X$ to contain a subfamily in $\mathcal O_p(X)$. The supplied record leaves $\mathcal O_p$, the admissible spaces and ultrafilters, and the cover-countability convention unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $800656$, exact source problem code catalogue entry; supplied complete source record. The registry attributes the problem to Tsaban, with proposed year 2003, and labels it open. This status is supplied metadata, not an independently verified mathematical assessment.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 29. Supplied source URL: https://arxiv.org/abs/math/0312182. The record states extraction method arxiv_latex and access date 2026-10-16; these are supplied provenance claims, not a claim of access during preparation of this draft.
• Supplied attribution/context labels: [GT1] and [GTpseq, Problem 0.10]. The record supplies no bibliographic expansions or separate URLs for these references.