Kourovka Notebook Problem 9.47

Definitions and mathematical statement

A topological space $K$ is scattered if every nonempty subset $A\subseteq K$, with its subspace topology, has an isolated point. For a topological group $G$, write \[ \mathcal S_{\mathrm{nc}}(G)=\{H\leq G: H\text{ is closed in }G\text{ and is not compact}\}. \] The source asks whether every scattered compact space $K$ admits a topological embedding \[ e:K\longrightarrow \left(\mathcal S_{\mathrm{nc}}(G),\tau_E\right) \] for some locally compact group $G$, which may depend on $K$. Here a topological embedding is a continuous injection that is a homeomorphism onto its image with the subspace topology, and $\tau_E$ denotes the source's $E$-topology.

The supplied record does not define the $E$-topology, so the topology on the target remains unresolved. It also does not explicitly state the separation conventions for "compact" or "locally compact group," including whether Hausdorffness is required. These conventions must be recovered from the source before this becomes a fully specified mathematical statement.

Short English statement

Does every scattered compact space occur, up to homeomorphism, as a subspace of the space of closed noncompact subgroups of some locally compact group, equipped with the source's $E$-topology? The definition of that topology and the separation conventions remain unresolved.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $700580$, exact source code catalogue entry, title "Kourovka Notebook Problem 9.47." Attribution: I. V. Protasov, proposed in 1984.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), Problem 9.47; also identified in the supplied record with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied background reports an explicitly unverified AI claim of a negative answer, attributed to unverified AI solutions: M. Kinyon and J. Urban, preprint of 17 September 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.
• The same explicitly unverified claim also cites Achyuth Jayadevan, preprint of 18 September 2026, https://zenodo.org/records/22838025.
• Supplied status curation dated 2026-10-16 retains the canonical label "open," classifies the evidence as unrefereed_claim, and reports a $\#$ marker for an unverified AI solution claim in the October 2026 21st issue. This is supplied dated curation, not an independently verified literature assessment. Its status-review quotation is truncated; the background supplies the complete links reproduced above.