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A metric space with its transfinite asymptotic dimension omega + 1

T0 review · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A concrete metric space is constructed whose transfinite and complementary-finite asymptotic dimensions are both omega+1, giving a counterexample to the omega conjecture.

arxiv 1908.00434 v2 pith:O4FAJDJL submitted 2019-08-01 math.MG math.AT

Yan Wu , Jingming Zhu This is my paper · ORCID
classification math.MGmath.AT
keywords omegaasymptoticdimensionmetricspacetransfinitecomplementary-finiteconjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Roughly, the 'asymptotic dimension' of a space measures how many different directions you need to cut the space into uniformly bounded pieces when you zoom out to infinity. It is usually a finite number, but Gromov's idea can be extended to infinite ordinal numbers, just as counting can go past the finite numbers to omega and beyond. The omega conjecture said that if a space's transfinite asymptotic dimension is infinite but finite in the ordinal sense, it must be exactly omega. This paper builds a space made of an infinite stack of grids, each one dimension higher than the last, placed further and further apart. The authors show that one can cover all but a finite part of the stack with two well-behaved families of pieces, which gives an upper bound of omega+1. They then try to show that omega is not enough, using a classical covering theorem about cubes. The delicate step in their lower-bound proof is the assertion that a certain connected piece of the cube contains a path between opposite faces; connectedness does not generally imply the existence of such a path. As written, the proof has a gap, though it may be repairable with a more careful argument along the grid skeleton.
Extended reading notes

Core claim

The paper claims to prove that the omega conjecture is false by constructing a metric space X_{omega+1} with trasdim(X_{omega+1}) = coasdim(X_{omega+1}) = omega+1; see the abstract, Proposition 3.1, and Proposition 3.2.

Load-bearing premise

The proof of Proposition 3.1 assumes that the connected component of the closed set ~U_0 produced by Karasev's theorem contains a path alpha joining opposite facets of the cube. Connected closed subsets of a cube need not be path connected (the topologist's sine curve is a counterexample), and the paper does not justify this step. The lower bound trasdim(X_{omega+1}) > omega depends directly on the existence of such a path.

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Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard covering theorems and on two lemmas from the authors' earlier paper. No fitted parameters or invented entities are introduced.

assumptions (3)
  • standard math Karasev's covering theorem (Theorem 3.1)
    Used to find a connected component of the closed cover that intersects opposite facets of the cube. The paper cites reference [8] and does not prove it.
  • domain assumption Lemma 3.1 from reference [4] (characterization of trasdim <= omega + l)
    Used in Proposition 3.1 to translate trasdim <= omega into the existence of bounded disjoint covers. Proven in the authors' prior paper and not re-proven here.
  • domain assumption Lemma 3.2 from reference [4] (coasdim <= omega + k implies trasdim <= omega + k)
    Used in Proposition 3.2 and Remark 3.2 to transfer the coasdim bound to trasdim and to rule out coasdim = omega.

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Pith. "Pith review of A metric space with its transfinite asymptotic dimension omega + 1." pith.science (2026). https://pith.science/paper/O4FAJDJL

@misc{pith2026190800434,
  author       = {Pith},
  title        = {Pith review of: A metric space with its transfinite asymptotic dimension omega + 1},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O4FAJDJL}},
  note         = {Machine review of arXiv:1908.00434}
}
read the original abstract

We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both omega+1, where omega is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not true.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. APD profiles and transfinite asymptotic dimension

    math.MG 2019-08 conditional novelty 7.0 of 10

    An infinity-pseudometric space has transfinite asymptotic dimension at most omega+n exactly when it admits a two-term integral APD profile of width n+1.

Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages · cited by 1 Pith paper

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    work page 2018

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