Selection Principles Milestone Problems — Problem 27 ([GJ, Question 2.4])

Definitions and mathematical statement

Let $X$ be a topological space and let $p$ be an ultrafilter on ${\mathbb{N}}$. Thus $p$ is a proper filter on ${\mathbb{N}}$ such that, for every $A\subseteq{\mathbb{N}}$, either $A\in p$ or ${\mathbb{N}}\setminus A\in p$. The supplied record does not specify whether $p$ must be nonprincipal.

Following the supplied definition, an open cover $\mathcal U$ of $X$ is an open $\gamma_p$-cover if it admits an enumeration $\mathcal U=\{U_n:n\in{\mathbb{N}}\}$ satisfying \[ \{n\in{\mathbb{N}}:x\in U_n\}\in p \qquad\text{for every }x\in X. \] Write $\mathcal O_p(X)$ for the collection of these covers. Membership is expressed by the existence of a suitable enumeration; the record does not specify whether repetitions are allowed or whether members equal to $X$ are forbidden.

Under the usual meaning of an open $\omega$-cover, $\Omega(X)$ consists of open covers $\mathcal U$ such that $X\notin\mathcal U$ and \[ (\forall F\subseteq X\text{ finite})(\exists U\in\mathcal U)\ F\subseteq U. \] This convention makes explicit notation left undefined in the supplied excerpt. Whether $\Omega(X)$ is additionally restricted to countable covers must be checked against the source conventions.

The selection property \[ P_p(X):\quad \binom{\Omega}{\mathcal O_p} \] means that every $\mathcal U\in\Omega(X)$ contains a subfamily $\mathcal V\subseteq\mathcal U$ belonging to $\mathcal O_p(X)$. The record calls this the $\gamma_p$-property, with reference label [GTpseq].

Question (supplied Problem 27). Is $P_p$ additive for each ultrafilter $p$? Under the interpretation that additivity means preservation under finite unions of subspaces, the requested assertion is \[ (\forall p)(\forall Z)(\forall A,B\subseteq Z)\quad \left(P_p(A)\land P_p(B)\right)\Longrightarrow P_p(A\cup B), \] where $p$ ranges over the intended ultrafilters on ${\mathbb{N}}$, $Z$ ranges over the intended ambient topological spaces, and $A$, $B$, and $A\cup B$ carry their subspace topologies. The same $p$ is used throughout each implication. This is a conditional formulation: the supplied record does not define additivity, specify the class of spaces, or settle the cover and ultrafilter conventions above. If countable unions are intended, the conclusion instead concerns $P_p(\bigcup_{n\in{\mathbb{N}}}X_n)$ whenever every $P_p(X_n)$ holds; that is a distinct question.

Short English statement

For a fixed ultrafilter $p$, the $\gamma_p$-property asks that every open $\omega$-cover contain an enumerated subcover whose membership indices belong to $p$ at every point. The problem asks whether this property is additive. A precise intended statement requires confirmation of the union operation, the class of spaces, and the cover and ultrafilter conventions. The registry supplies the label "open"; no independent status verification is asserted.

Sources

• Local UnsolvedMath registry, permanent ID $800654$, source code catalogue entry, title "Selection Principles Milestone Problems --- Problem 27 ([GJ, Question 2.4])." The supplied record lists Tsaban as proposer, gives the year 2003, and labels the problem open.
• Supplied source list: Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 27. Supplied 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 independently verified access claims.
• Supplied attribution/context: [GJ, Question 2.4]. The supplied record does not expand this reference label.
• The supplied statement attributes the terminology "$\gamma_p$-property" to [GTpseq]. The supplied record does not provide bibliographic details for this reference label.