Kourovka Notebook Problem 9.38

Definitions and mathematical statement

For a topological group $G$, give each subgroup $H\leq G$ the subspace topology. Say that $H$ is compactly covered if \[ H=\bigcup\{K\leq H:K\text{ is compact}\}. \] Equivalently, every element of $H$ belongs to some compact subgroup of $H$. Such a subgroup $H$ is maximal compactly covered in $G$ if, for every compactly covered subgroup $L\leq G$, \[ H\leq L\quad\Longrightarrow\quad L=H. \] Maximality here is with respect to inclusion among all compactly covered subgroups, without assuming that they are closed.

Question (Yu. N. Mukhin). Let $G$ be a null-dimensional locally compact group. Must every maximal compactly covered subgroup $H\leq G$ satisfy \[ \overline{H}^{\,G}=H? \] The source does not define "null-dimensional" or explicitly state separation conventions. With the conventional reading that $G$ is Hausdorff and that null-dimensional means having a base of sets that are both open and closed, this asks whether the displayed assertion holds for every such $G$ and every such $H$. These conventions require confirmation against the source; no countability or metrizability hypothesis is supplied.

Short English statement

Yu. N. Mukhin asks whether every subgroup maximal among those that are unions of compact subgroups is closed in a null-dimensional locally compact group. The meaning of null-dimensional and the intended separation conventions require source confirmation. The supplied registry labels this problem open; that label is not an independently verified mathematical assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $700571$, exact source problem code catalogue entry, title Kourovka Notebook Problem 9.38. The record attributes the problem to Yu. N. Mukhin, proposed in 1984, and labels its status open.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), Problem 9.38; also identified with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic attribution is supplied evidence and has not been independently checked.