VibeMathedMath problems solved with AI
Sign in

Compact-range Vietoris powers: from ω+1\omega+1 to all countable ordinals above ω\omega

Question 1 on page 11 of Christopher Caruvana and Jared Holshouser, An Adaptation of the Vietoris Topology for Ordered Compact Sets (arXiv:2507.17936v3), asks whether K(α,ord)\mathbb K(\alpha,\mathrm{ord}) is σ\sigma-compact or Lindelöf for ω<α<ω1\omega<\alpha<\omega_1, explicitly singling out α=ω+1\alpha=\omega+1. Here the carrier consists of arbitrary functions with actual compact image, equipped with the paper’s Vietoris-power topology; for countable ordinals the default index is ω\omega.

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI co-developed
Method
Argument
Field
General topology; covering and selection properties
Posed by
Christopher Caruvana and Jared Holshouser, Question 1, p. 11, An Adaptation of the Vietoris Topology for Ordered Compact Sets, arXiv:2507.17936v3
Year posed
2025
Years open
1y
Solved
2026
Model
ChatGPT Astra
Vendor
OpenAI
Collaborators
Lucas Alves de Almeida
Verification
Lean-checked, statement unaudited
Publication
Announced
Significance
12 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

First, On the Vietoris Power of a Convergent Sequence settles the highlighted case α=ω+1\alpha=\omega+1: the space is second-countable and Lindelöf, but not σ\sigma-compact. Compact families of surjections have finite bounds on where each isolated value occurs; a delayed diagonal defeats every countable compact cover.

The stronger manuscript proves, for every ω<α<ω1\omega<\alpha<\omega_1, second countability and Lindelöfness, together with failure of both σ\sigma-compactness and the Menger property. It combines a countable-base argument, transfer through a closed copy of [0,ω][0,\omega], and a direct fixed-compact-image diagonal. Explicit increasing open covers and the same diagonal defeat all finite selections, proving non-Mengerness.

Thus the original question has a positive answer for Lindelöfness and a negative answer for σ\sigma-compactness. “Proved” refers to this classification theorem.

What the AI did

According to the manuscripts’ AI disclosures, ChatGPT Astra contributed substantive new mathematics, as well as the Lean formalization and manuscript preparation, under Lucas Alves de Almeida’s direction. In the first stage, concerning ω+1\omega+1, Astra performed almost all of the mathematical development, formalization and preparation, with Lucas checking and editing. In the stronger countable-ordinal work, Lucas proposed and guided the extension, directed the delayed-diagonal investigation, and requested the closed-subspace presentation; Astra developed the main general argument and performed the primary mathematical development, formalization, verification and manuscript preparation. The methods include a countable-base argument, finite coordinate bounds on compact families, a delayed diagonal, transfer through closed embeddings, and explicit open covers witnessing failure of the Menger property. The AI contribution was proof development, not merely formalization of an already known human result. “AI co-developed” is selected conservatively to reflect the human-directed workflow.

Verification

Audited here on 30 September 2026 by reading the repository at the reviewed snapshot d2e73e4. Lean 4.19.0 with mathlib pinned at c44e0c8e. The principal theorem is VietorisOrdinals.ordinal_full (a : Ordinal) (hlo : omega < a) (hhi : a < omega_1), concluding MainClaim (OrdinalSpace a) together with the failure of the Menger property, and the hypotheses are exactly the range the question asks about with nothing extra. The fidelity check that matters was done by the repository itself and was confirmed here: FullAudit.lean restates the three conclusions - second countable, Lindelof, not sigma-compact - in a bare `example` that does not pass through the MainClaim abbreviation, so a definition quietly shadowing the claim would show up. Lean-checked rather than Lean-verified: the site did not rebuild the project, the submitter reports that a full package build was not completed locally because of resource limits, and the statement is anchored only by the author's own repository. The submitter's note that a further AI review is additional AI checking and not independent human endorsement is correct and is why this stays a Candidate.

Sources

Submitted by StormyGander827 on

Changelog2 changes

Discussion