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REVIEW 3 major objections 6 minor 17 references

$\mathbb{M}^*$, $\mathbb{N}^*$, and $\mathbb{H}^*$

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Under CH, every self-homeomorphism of N* lifts to M*, and the half-line remainder gains an order-reversing self-map.

desk verdict A genuinely new CH lifting theorem for M* over N* with a careful recursion; Theorem 2 is a nice corollary but leans on an unproved conjugacy result from a coauthor's preprint. read the letter →

arxiv 2505.04425 v1 pith:MYI25JGB submitted 2025-05-07 math.GN math.LO

Will Brian , Alan Dow , Klaas Pieter Hart This is my paper · ORCID
classification math.GNmath.LO MSC 54D4003E3506D5003C20
keywords Čech-StoneremaindersautohomeomorphismsContinuumHypothesislatticebaseelementarityliftingpropertyorder-reversinghomeomorphism
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

This paper proves that, assuming the continuum hypothesis (CH), every autohomeomorphism of $\mathbb N^*$ — the Čech-Stone remainder of the natural numbers — can be lifted through the natural projection to an autohomeomorphism of $\mathbb M^*$, the remainder of $\mathbb N\times[0,1]$. A separate consistency result shows this lifting property can fail in some models of ZFC, so the two results together answer an open question about when the homeomorphism groups of these remainders are related by the projection. The paper also derives, as a corollary, that CH gives an order-reversing autohomeomorphism of $\mathbb H^*$, the remainder of the half-line $[0,\infty)$, a feature that is independent of ZFC and cannot be produced by trivial homeomorphisms. The interest is that this is a rare statement about Čech-Stone remainders that follows both from forcing axioms and from CH, but not from ZFC alone.

What carries the argument

The central object is the reduced power $B^{\mathbb N}/\mathrm{fin}$, where $B$ is a countable distributive lattice base for the closed subsets of $[0,1]$ (say the lattice generated by closed intervals with rational endpoints). This reduced power is a base for the closed sets of $\mathbb M^*$, and automorphisms of its partial order are dual to autohomeomorphisms of $\mathbb M^*$. The proof of Theorem 1 uses CH to build, by recursion of length $\omega_1$, a partial map $\varphi:B^{\mathbb N}\to B^{\mathbb N}$ whose induced map on $B^{\mathbb N}/\mathrm{fin}$ is an automorphism; at each step, elementary equivalence and saturation of the ultrapower $B^{\mathbb N}/u$ let the construction 'look ahead' to later coordinates and decide all lattice formulas on a tail. The induced $H$ satisfies $\pi^*\circ H = h\circ \pi^*$. For Theorem 2, the quotient $\mathbb M^*/\sim$ that glues the right end of each fiber $I_u$ to the left end of $I_{\sigma(u)}$ is $\mathbb H^*$, and composing an order-preserving lift with the flip on each interval component gives an order-reversing homeomorphism that respects $\sim$.

What would settle it

Look for a model of ZFC + CH in which some autohomeomorphism $h$ of $\mathbb N^*$ fails to lift, i.e., no $H$ on $\mathbb M^*$ satisfies $\pi^*\circ H = h\circ\pi^*$; such a model would refute Theorem 1. Alternatively, test the cited conjugacy theorem that $\sigma$ and $\sigma^{-1}$ are conjugate under CH: since this paper does not re-prove it, a counterexample or a gap there would leave Theorem 2's gluing step unsupported.

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Extended reading notes

Core claim

The central claim is Theorem 1: assuming CH, for every autohomeomorphism $h$ of $\mathbb N^*$ there is an autohomeomorphism $H$ of $\mathbb M^*$ such that $\pi^*\circ H = h\circ \pi^*$, where $\pi^*:\mathbb M^*\to\mathbb N^*$ is induced by the natural projection $(n,x)\mapsto n$. Theorem 2 states that CH also implies the existence of an order-reversing autohomeomorphism of $\mathbb H^*$. Theorem 2 is a corollary of Theorem 1 together with the cited recent theorem that under CH the shift map $\sigma$ on $\mathbb N^*$ and its inverse $\sigma^{-1}$ are conjugate; the proof composes an order-preserving lift $F$ with the flip on each interval fiber and checks that the resulting map respects the equivalence relation whose quotient is $\mathbb H^*$.

Load-bearing premise

The load-bearing premise is the cited, unproved-here theorem that under CH the shift map $\sigma$ on $\mathbb N^*$ is conjugate to its inverse; if that theorem is false, the order-reversing homeomorphism of $\mathbb H^*$ does not follow.

Editorial extensions

If this is right

  • Under CH, the projection $\pi^*$ has the lifting property for every autohomeomorphism of $\mathbb N^*$, so the autohomeomorphism group of $\mathbb M^*$ maps onto that of $\mathbb N^*$.
  • Because OCA_T makes all autohomeomorphisms of $\mathbb N^*$ trivial, the lifting property also follows from OCA_T, but it fails in some ZFC model, so it is independent of ZFC.
  • Under CH, $\mathbb H^*$ has a nontrivial autohomeomorphism that permutes its standard subcontinua and reverses the quasiorder on each one; trivial autohomeomorphisms of $\mathbb H^*$ are always order-preserving.
  • The existence of an order-reversing autohomeomorphism of $\mathbb H^*$ is independent of ZFC: it follows from CH, while OCA_T+MA implies all such autohomeomorphisms are trivial and hence order-preserving.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof of Theorem 1 only needs a countable distributive lattice base for the fibers, so the same lifting construction should work for $\mathbb N\times K$ for any compact space $K$ with such a base; checking that would show whether the unit interval is essential or merely convenient.
  • Theorem 2 is really a corollary of two ingredients: the lifting property and the conjugacy of the shift with its inverse. Any axiom that supplies both — not necessarily CH — would also produce an order-reversing autohomeomorphism of $\mathbb H^*$.
  • The domino picture in Remark 14 suggests a recipe for cutting-and-pasting a flipping homeomorphism on other remainders obtained as quotients of $\mathbb M^*$ by gluing fibers, which could be tested independently of the full CH recursion.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies the Čech-Stone remainders M* = (N×[0,1])* and N*. The main theorem (Theorem 1) states that, assuming CH, every autohomeomorphism of N* can be lifted to an autohomeomorphism of M* through the natural projection π*. The proof works with a countable distributive lattice base for the closed sets of the unit interval, identifies a base for the closed subsets of M* with the reduced power B^N/fin, and constructs an automorphism of the quotient lattice by a transfinite recursion of length ω1. The recursion uses a strengthened elementarity condition (*)_α and a saturation argument on ultrapowers to ensure that every lattice formula is preserved 'almost everywhere'. A second theorem (Theorem 2) derives, from Theorem 1 and a recent result of the first author on the conjugacy of the shift map on N* with its inverse under CH, the existence of an order-reversing autohomeomorphism of H* = [0,∞)*.

Significance. If correct, Theorem 1 is a significant contribution to the program on autohomeomorphisms of Čech-Stone remainders: it provides a lifting property that holds both under forcing axioms (via triviality of all autohomeomorphisms) and under CH, answering a question from Dow–Hart [4]. The proof method is novel and likely reusable: it combines lattice duality with model-theoretic elementarity and saturation in a way that is rare in this area. Theorem 2 settles the consistency of an order-reversing autohomeomorphism of H* from CH, complementing Vignati's OCA_T+MA theorem. The paper is generally well written, with clear motivation and a helpful 'look-ahead' explanation of why the naive recursion fails. The main gaps are in the detailed verification of the recursive construction, not in the overall strategy, and they appear to be readily fixable.

major comments (3)
  1. [Section 3, 'Other partitions' and 'Building D_α'] The construction treats D_u as a function on N (e.g., in the definition of D_α(k) on intervals [N_m,N_{m+1})), but D_u was chosen as an element of the ultrapower B_v = B^N/v, i.e., an equivalence class modulo v. The proof omits the necessary step of fixing a representative in B^N for each D_u and does not discuss how the definition of the sets B_{u,s} and of the final D_α depends on that choice. Since the verification of (*)_{α+1} is pointwise on intervals, this is load-bearing and should be made explicit.
  2. [Section 3, 'Other partitions'] The assertion that the family Q_m is pairwise disjoint after shrinking the sets B_{u,s} to be subsets of h+(A_s) is not justified. For a fixed s, different u ∈ F_s may have overlapping B_{u,s}; the fact that each B_{u,s} is contained in h+(A_s) does not prevent such overlaps. Because the definition of D_α on [N_m,N_{m+1}) relies on a unique pair (s,u) for each k, the authors need to prove that the B_{u,s} can be chosen pairwise disjoint (for example, by assigning each k to the first u in a fixed enumeration) while preserving the covering and membership properties used in the verification.
  3. [Section 4, Proof of Theorem 2] The proof relies on the main theorem of [1] (a preprint of the first author) that under CH the shift map σ and its inverse σ^{-1} are conjugate. This is an external result that is not proved or even stated in the paper. The dependence is acknowledged in the text, but since the theorem is load-bearing for the existence of the order-reversing homeomorphism of H*, the authors should state the precise theorem used and either prove it or clearly mark Theorem 2 as conditional on [1].
minor comments (6)
  1. [Section 3, condition (2)] In the initial list of conditions on φ, condition (2) defines A_2 = {k : C(k) ⊆ B(k)}, but in the recursion the same condition is written with A_2 = {k : C_γ(k) ⊆ C_β(k)} (and similarly for D). The orientation of the inclusion in the second conjunct appears to be reversed; please correct the typo.
  2. [Section 3, 'The construction'] The enumeration of B^N is denoted ⟨B_α : α<ω_1⟩, but B is also used for the countable lattice base. This is confusing at the point where 'let C_α be the first term of the sequence ⟨B_α⟩' appears; consider renaming the enumeration (e.g., ⟨E_α⟩).
  3. [Section 3, 'Other partitions'] The notation B^*_{u,s} is used in 'the family {B^*_{u,s} : u∈A^*_s} covers h[A^*_s]', which appears to be a typo for B_{u,s}; also 'F_u' in 'such that {B^*_{u,s} : s∈F_u}' should presumably be 'F_s'.
  4. [Section 3, 'Making D_α'] The phrase 'the number of free variables in γ_m' should read 'in χ_m'.
  5. [Section 4, Proposition 12] In the proof, f is a homeomorphism between co-compact subsets of H, so f(a_n) is undefined for finitely many n; the displayed definitions of f(¯a) and f(¯b) should be qualified as holding for all sufficiently large n.
  6. [References] Reference [7] gives a DOI that appears to belong to a different journal; please verify the DOI for Hart's survey.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Theorem 1 is proved from scratch; Theorem 2 is a transparently acknowledged corollary whose cited conjugacy theorem is an independent, parameter-free result.

full rationale

The claimed derivation chain is not circular. Theorem 1 is proved in Section 3 by an explicit CH recursion constructing a lattice automorphism of B^N/fin from the given autohomeomorphism h of N* via the induced map h+, and the verification of condition (*)_{alpha+1} is carried out internally. Lemma 8 then verifies pi*∘H = h∘pi* without assuming a lift exists. The only self-citation is the invocation of the first author's Theorem [1] in the proof of Theorem 2: 'By a recent theorem of the first author (the main theorem of [1]), CH implies sigma and sigma^{-1} are conjugate in the autohomeomorphism group of N*.' This is load-bearing—it supplies the permutation f needed for H = F∘flip*_N to respect the equivalence relation ∼ on M*, so that a well-defined map on H* ≅ M*/∼ is obtained. However, it is not circular: [1] is a separate earlier result with its own proof, its CH hypothesis does not include the target conclusion (an order-reversing autohomeomorphism of H*), and the paper explicitly describes Theorem 2 as 'a corollary' of Theorem 1 and [1]. No fitted parameters are introduced and no prediction is renamed as an input. The dependence on an unverified coauthor preprint is a correctness and verification risk, not a circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The main theorems are proved under ZFC + CH. Theorem 1's proof is self-contained modulo standard facts from model theory and Wallman duality. Theorem 2 additionally depends on Brian's recent conjugacy result [1], a load-bearing citation from a coauthor's preprint. No free parameters or invented entities are introduced.

assumptions (6)
  • domain assumption ZFC + Continuum Hypothesis (CH)
    Both theorems are stated and proved under CH. CH is used in Section 3 to enumerate B^N as ⟨B_α : α < ω1⟩ and to run the recursion of length ω1.
  • domain assumption Brian's conjugacy theorem (arXiv:2402.04358): under CH, the shift σ and σ^{-1} are conjugate in Aut(N*)
    Used in the proof of Theorem 2 (Section 4) to ensure the lifted homeomorphism H respects the gluing equivalence classes that form H*. This result is cited from a coauthor's preprint and is not re-proved here.
  • standard math Saturation of countably incomplete ultrapowers and extension of partial elementary maps (Hodges Lemma 10.1.3 / 8.1.3)
    In Section 3, the proof needs, for each u ∈ N*, an element D_u in B_v realizing the type determined by the partial map C_β ↦ D_β extended by C_α ↦ D. This relies on countable saturation of the ultrapower B_v.
  • standard math Wallman representation theorem for distributive lattices (Wallman 1938) and the identification of a lattice base for the closed sets of M*
    The proof describes H dually via an automorphism of the reduced power B^N/fin, relying on the correspondence between closed sets of M* and equivalence classes of B^N/fin (Section 3).
  • domain assumption The quotient space M*/∼ is homeomorphic to H* (Hart's survey [7, Theorem 2.4])
    Used in Section 4 to transfer the homeomorphism H of M* to an order-reversing autohomeomorphism h of H* by checking that H respects the equivalence relation that glues 1̄_u to 0̄_σ(u).
  • domain assumption Topological definability of the order on each component I_u (Proposition 10, citing [7, Section 2])
    Used in Section 4 to conclude that H is order-preserving when it fixes the 'lower half' K, since the endpoints 0̄_u and 1̄_u are topologically definable and the order is characterized by subcontinua.

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Cite this review

Pith. "Pith review of $\mathbb{M}^*$, $\mathbb{N}^*$, and $\mathbb{H}^*$." pith.science (2026). https://pith.science/paper/MYI25JGB

@misc{pith2026250504425,
  author       = {Pith},
  title        = {Pith review of: $\mathbbM^*$, $\mathbbN^*$, and $\mathbbH^*$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MYI25JGB}},
  note         = {Machine review of arXiv:2505.04425}
}
abstract

Let $\mathbb{M} = \mathbb N \times [0,1]$. The natural projection $\pi: \mathbb{M} \rightarrow \mathbb N$, which sends $(n,x)$ to $n$, induces a projection mapping $\pi^*: \mathbb{M}^* \rightarrow \mathbb N^*$, where $\mathbb{M}^*$ and $\mathbb N^*$ denote the \v{C}ech-Stone remainders of $\mathbb{M}$ and $\mathbb N$, respectively. We show that $\mathsf{CH}$ implies every autohomeomorphism of $\mathbb N^*$ lifts through the natural projection to an autohomeomorphism of $\mathbb{M}^*$. That is, for every homeomorphism $h: \mathbb N^* \rightarrow \mathbb N^*$ there is a homeomorphism $H: \mathbb{M}^* \rightarrow \mathbb{M}^*$ such that $\pi^* \circ H = h \circ \pi^*$. This complements a recent result of the second author, who showed that this lifting property is not a consequence of $\mathsf{ZFC}$. Combining this lifting theorem with a recent result of the first author, we also prove that $\mathsf{CH}$ implies there is an order-reversing autohomeomorphism of~$\mathbb H^*$, the \v{C}ech-Stone remainder of the half line $\mathbb H = [0,\infty)$.

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Works this paper leans on

17 extracted references · 13 canonical work pages

  1. [1]

    W. R. Brian,DoesP(ω)/finknow its right hand from its left?, posted on 2 May 2024, DOI 10.48550/arXiv.2402.04358

    work page doi:10.48550/arxiv.2402.04358 2024

  2. [4]

    Dow and K

    A. Dow and K. P. Hart, ˇCech-Stone remainders of spaces that look like[0,∞), Acta Univ. Carolin. Math. Phys.34(1993), no. 2, 31–39. Selected papers from the 21st Winter School on Abstract Analysis (Podˇ ebrady, 1993). MR1282963

    work page 1993

  3. [2]

    De Bondt, I

    B. De Bondt, I. Farah, and A. Vignati,Trivial isomorphisms between reduced products, posted on 28 October 2024, DOI 10.48550/arXiv.2307.06731. To appear inIsrael Journal of Math- ematics

    arXiv doi:10.48550/arxiv.2307.06731 2024

  4. [3]

    Dow,Autohomeomorphisms of pre-images ofN ∗, Topology Appl.368(2025), Paper No

    A. Dow,Autohomeomorphisms of pre-images ofN ∗, Topology Appl.368(2025), Paper No. 109348, 14 pages, DOI 10.1016/j.topol.2025.109348. 14 WILL BRIAN, ALAN DOW, AND KLAAS PIETER HART

    arXiv 2025

  5. [5]

    Alan Dow and Klaas Pieter Hart,Cut points in ˇCech-Stone remainders, Proc. Amer. Math. Soc.123(1995), no. 3, 909–917, DOI 10.2307/2160818. MR1216810

    work page doi:10.2307/2160818 1995

  6. [6]

    Gelfand and A

    I. Gelfand and A. Kolmogoroff,On rings of continuous functions on topological spaces, C. R. (Dokl.) Acad. Sci. URSS, n. Ser.22(1939), 11–15 (English). Zbl 0021.41103

    work page arXiv 1939

  7. [7]

    317–352, DOI 10.1016/0887- 2333(92)90021-I

    Klaas Pieter Hart,The ˇCech-Stone compactification of the real line, Recent progress in general topology (Prague, 1991), North-Holland, Amsterdam, 1992, pp. 317–352, DOI 10.1016/0887- 2333(92)90021-I. MR1229130

    work page doi:10.1016/0887- 1991

  8. [8]

    42, Cambridge University Press, Cambridge, 1993

    Wilfrid Hodges,Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR1221741

    1993

Show all 17 references
  1. [9]

    MR1462612

    ,A shorter model theory, Cambridge University Press, Cambridge, 1997. MR1462612

    1997

  2. [10]

    Pure Appl

    Justin Tatch Moore,Some remarks on the Open Coloring Axiom, Ann. Pure Appl. Logic 172(2021), no. 5, Paper No. 102912, 6, DOI 10.1016/j.apal.2020.102912. MR4228344

    2021

  3. [11]

    J.23(1956), 409–419

    Walter Rudin,Homogeneity problems in the theory of ˇCech compactifications, Duke Math. J.23(1956), 409–419. MR0080902

    1956

  4. [12]

    940, Springer-Verlag, Berlin-New York, 1982

    Saharon Shelah,Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR0675955

    1982

  5. [13]

    Saharon Shelah and Juris Stepr¯ ans,PFA implies all automorphisms are trivial, Proc. Amer. Math. Soc.104(1988), no. 4, 1220–1225, DOI 10.2307/2047617. MR0935111

    1988 doi

  6. [14]

    84, Amer- ican Mathematical Society, Providence, RI, 1989

    Stevo Todorˇ cevi´ c,Partition problems in topology, Contemporary Mathematics, vol. 84, Amer- ican Mathematical Society, Providence, RI, 1989. MR0980949

    1989

  7. [15]

    1, 1–13, DOI 10.1016/0166-8641(93)90127-Y

    Boban Veli˘ ckovi´ c, OCAand automorphisms ofP(ω)/fin, Topology Appl.49(1993), no. 1, 1–13, DOI 10.1016/0166-8641(93)90127-Y. MR1202874

    1993 doi

  8. [16]

    Alessandro Vignati,Rigidity conjectures for continuous quotients, Ann. Sci. ´Ec. Norm. Sup´ er. (4)55(2022), no. 6, 1687–1738 (English, with English and French summaries). MR4517685

    2022

  9. [17]

    Henry Wallman,Lattices and topological spaces, Ann. of Math. (2)39(1938), no. 1, 112–126, DOI 10.2307/1968717. MR1503392 W. R. Brian, Department of Mathematics and Statistics, University of North Car- olina at Charlotte, Charlotte, NC, USA Email address:wbrian.math@gmail.com U...

    1938 doi

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