Spectrality of Infinite Symmetric Orbit-Space Quotients

Definitions and mathematical statement

For a topological space $X$, define $x\sim y$ if, for every open subset $U\subseteq X$, one has $x\in U$ if and only if $y\in U$. Its Kolmogorov quotient is $\operatorname{KQ}(X)=X/{\sim}$, endowed with the quotient topology.

A topological space $Y$ is spectral if it is $T_0$, is quasi-compact, has a basis of quasi-compact open subsets closed under finite intersections, and every nonempty irreducible closed subset has a unique generic point. Here quasi-compact means that every open cover has a finite subcover; a nonempty closed subset is irreducible if it is not the union of two proper closed subsets; and a generic point of a closed subset $C$ is a point $y$ with $\overline{\{y\}}=C$.

The supplied question is \[ \text{Is }\operatorname{KQ}\!\left(A_{\mathbb K,\infty}^{n}/G\right)\text{ a spectral space?} \] This question is retained with the intended objects and topology of the cited source, which are not defined in the supplied record. In particular, the record does not specify the allowed $\mathbb K$, the range of $n$, the meaning of $A_{\mathbb K,\infty}^{n}$, the group $G$ or its action, or the topology on $A_{\mathbb K,\infty}^{n}/G$. Consequently, the displayed question is a source-dependent formulation: its parameter quantifiers and hypotheses cannot be made explicit from the supplied evidence.

Short English statement

The question asks whether the space obtained from $A_{\mathbb K,\infty}^{n}/G$ by identifying topologically indistinguishable points is spectral. The underlying objects, topology, and parameter restrictions are missing from the supplied record, so a fully specified statement requires source clarification. The supplied status is open; this draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005476$, source problem code catalogue entry; title: Spectrality of Infinite Symmetric Orbit-Space Quotients. The record gives the proposed year as 2023 and does not identify a proposer.
• Supplied report citation: New Directions in Real Algebraic Geometry (2023), Oberwolfach Reports, pp. 813--881, DOI 10.4171/owr/2023/15. https://doi.org/10.4171/owr/2023/15. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Kummer--Riener, Journal of Algebra, DOI 10.1016/j.jalgebra.2024.11.016. https://doi.org/10.1016/j.jalgebra.2024.11.016. The supplied curation describes this as a 2025 primary paper; no paper title or exact conjecture location is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that the 2025 primary paper still presents the Kolmogorov-quotient spectral-space assertion as a conjecture, and reports that no resolving proof or counterexample was found. This is supplied dated curation derived from the OWR record, not independently verified literature assessment. Its listed sources are the Kummer--Riener paper and the OWR report linked above; the record explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official Oberwolfach report. These are supplied verification claims, not verification performed for this draft.