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REVIEW 3 major objections 6 minor 8 references

Cardinal Sequences of Lindel\"of scattered P-spaces

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that every length-$\omega_2$ cardinal sequence of an LLSP space satisfies a necessary inequality, that any sequence satisfying a stronger inequality is realized, and that the two inequalities are consistently distinct.

desk verdict Serious paper, but the stress-test is right: Lemma 5.4's amalgamation misses overlaps, so Theorem 5.1 as written does not follow. read the letter →

arxiv 2411.17791 v1 pith:WP2VSVXD submitted 2024-11-26 math.GN

J.C Martínez , L. Soukup This is my paper · ORCID
classification math.GN MSC 54A2554G1203E3554D20
keywords cardinalsequencesLLSPspacesscatteredP-spacesLindelöfforcingconesystemsCantor-Bendixsonlevels
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

The paper studies the cardinal sequences of locally Lindelöf, scattered, Hausdorff P-spaces (LLSP spaces), i.e. the list of sizes of successive Cantor–Bendixson levels. Its central result, Theorem 4.4, gives a sufficient condition and a separate necessary condition for a length-$\omega_2$ sequence to be such a cardinal sequence, expressed through the cardinal invariants $\hat A$ and $\hat I$. Because the two conditions are consistently different, the class $\mathrm{CP}(\omega_2)$ lies between them, and the natural transfer of the known LCS characterization to LLSP spaces is not a theorem of ZFC. The paper also shows that, under $2^{\omega_1}=\omega_2$, it is consistent that $\langle\omega_2\rangle^\omega \frown \langle\omega_3\rangle$ is the cardinal sequence of an LLSP space of height $\omega+1$, so the usual bound $\lambda\le \kappa^{\omega_1}$ can fail at limit levels of cofinality $\omega$.

What carries the argument

The engine of the paper is the notion of an LLSP-cone assignment: a system of neighborhoods $U$ on a partitioned set $X$, together with a countable $D$-function, satisfying conditions (a)–(d) of Definition 2.3. Theorem 2.6 shows that LLSP spaces are exactly the spaces arising from such assignments, so constructing LLSP spaces becomes a combinatorial problem about these $U,D$ systems. The invariants $\hat A(\kappa,\lambda)$ and $\hat I(\kappa,\lambda)$ measure the cost of finding large almost-disjoint families and of covering a set by countably many smaller sets; they supply the bounds in Theorem 4.4. For the height $\omega+1$ example, the paper defines a partial order $P_f$ whose conditions are partial approximations to an LLSP-cone assignment, with dense conditions forcing each point at level $\beta$ to meet every lower level in a cofinal set, and proves that this poset is $\omega_2$-closed and has the $\omega_3$-chain condition.

What would settle it

Find a single ordinal $\alpha<\omega_3$ for which no LLSP space with constant cardinal sequence $\langle\omega_1\rangle^\alpha$ exists; then the sufficient condition in Theorem 4.4(1) fails as a construction, because the new points are attached to exactly those spaces. A direct place to look is $\alpha=\omega_2$, where the earlier paper's Theorem 2.1 asserts existence of a space with $\mathrm{CS}=\langle\omega_1\rangle^{\omega_2}$.

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Extended reading notes

Core claim

The central discovery is a sandwich for $\mathrm{CP}(\omega_2)$. Theorem 4.4(1) states that if $\langle\kappa_\alpha:\alpha<\omega_2\rangle$ is a sequence of uncountable cardinals with $\kappa_\beta<\hat A(\kappa_\alpha,\omega_1)$ for every $\alpha<\beta<\omega_2$, then the sequence is the cardinal sequence of an LLSP space. Theorem 4.4(2) states that if such a sequence is realized, then $\kappa_\beta<\hat I(\kappa_\alpha,\kappa_\alpha)$ for every $\alpha<\beta<\omega_2$. Since $\hat A(\kappa,\lambda)\le \hat I(\kappa,\lambda)$ when $\operatorname{cf}(\lambda)>\omega$, the sufficient condition is stronger, and the two bounds are consistently distinct: in one model $\omega_3=\hat A(\omega_1,\omega_1)<\hat I(\omega_1,\omega_1)=\omega_4$. The paper also constructs, by forcing, an LLSP space with cardinal sequence $\langle\omega_2\rangle^\omega \frown \langle\omega_3\rangle$ under $2^{\omega_1}=\omega_2$, which demonstrates that the naive analogue of the classical characterization fails and that the converse of the sufficient condition fails in that model.

Load-bearing premise

The construction in Theorem 4.4(1) and the final example glue copies of LLSP spaces with constant cardinal sequence $\langle\omega_1\rangle^\alpha$, imported from the authors' earlier paper [8]; if those base spaces do not exist for every $\alpha<\omega_3$, the new construction has nothing to build on.

Editorial extensions

If this is right

  • Every LLSP space of height $\omega_2$ has a cardinal sequence satisfying $\kappa_\beta<\hat I(\kappa_\alpha,\kappa_\alpha)$ for all $\alpha<\beta<\omega_2$.
  • Any length-$\omega_2$ sequence of uncountable cardinals with $\kappa_\beta<\hat A(\kappa_\alpha,\omega_1)$ for all $\alpha<\beta<\omega_2$ is realized by an LLSP space.
  • The two bounds in Theorem 4.4 are consistently different: $\hat A(\omega_1,\omega_1)=\omega_3<\hat I(\omega_1,\omega_1)=\omega_4$ in one forcing extension.
  • It is consistent with $2^{\omega_1}=\omega_2$ that $\langle\omega_2\rangle^\omega \frown \langle\omega_3\rangle$ belongs to $\mathrm{CP}(\omega+1)$, so the bound $s(\delta)\le (\prod_{\gamma\in C}s(\gamma))^{\omega_1}$ fails at limits of cofinality $\omega$.
  • The naive transfer of the classical LCS characterization to LLSP spaces is false in ZFC: in a c.c.c. forcing extension with $2^\omega=2^{\omega_1}=\omega_3$, the pair $\langle\omega_1,2^{\omega_1}\rangle$ is not an LLSP cardinal sequence.

Reading between the lines

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

  • Editorial extension: the forcing construction of Theorem 5.1 may generalize to other pairs $\langle\lambda\rangle^\omega \frown \langle\mu\rangle$, suggesting that limit levels of cofinality $\omega$ can host large cardinals whenever the two invariants allow it; the paper does not test this.
  • Editorial extension: the gap between $\hat A(\kappa,\omega_1)$ and $\hat I(\kappa,\kappa)$ for cardinals such as $\kappa=\omega_2$ is left open; computing these invariants in more models could determine whether a single condition can replace the two bounds in Theorem 4.4.
  • Editorial extension: the text cites a reduction theorem and the free-subset theorem with placeholder marks ($[?]$); those references need to be pinned down before the proof is fully checkable.
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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 cardinal sequences of locally Lindelöf, scattered, Hausdorff P-spaces (LLSP spaces). It introduces weak LLSP-cone assignments and shows (Theorem 2.6) that such assignments correspond exactly to LLSP spaces, providing a uniform construction method. Section 3 establishes basic bounds on LLSP cardinal sequences, including a decomposition into finitely many decreasing constant blocks and bounds of the form kappa^{omega_1}. Section 4 defines two cardinal functions, A-hat(kappa,lambda) and I-hat(kappa,lambda), and proves that a length-omega_2 sequence is realizable provided kappa_beta < A-hat(kappa_alpha,omega_1) for all alpha<beta, while every realizable sequence must satisfy kappa_beta < I-hat(kappa_alpha,kappa_alpha). Section 5 presents a forcing construction intended to show it is consistent that 2^{omega_1}=omega_2 and <omega_2>^omega^<omega_3> belongs to CP(omega+1), which would separate the two bounds and refute a naive transfer of the Juhász-Weiss characterization.

Significance. If the proofs are correct, the paper makes a substantial contribution: it gives a flexible cone-system framework for constructing LLSP spaces and shows that the class CP(omega_2) is sandwiched between two distinct combinatorial conditions, with a consistency result demonstrating that the naive LLSP analogue of the Juhász-Weiss theorem fails. The definitions and neighborhood-base verifications are explicit and checkable, and the paper is honest about its reliance on the authors' prior constructions in [8]. However, two load-bearing proofs, Lemma 4.10 and especially Lemma 5.4, currently have gaps that must be repaired before the main claims are established.

major comments (3)
  1. [Lemma 5.4, Claim, Case 3] The displayed equality U_r(s) cap U_r(t) = (V_xi \ {y_xi}) cup {w^xi_n : n<omega} is not justified. Since U_r(y_xi)=U_nu(y_xi) cup {w^xi_n} and U_r(bar y_xi)=U_mu(bar y_xi) cup {w^xi_n}, and since X_nu cap X_mu = X^*, the intersection contains U_nu(y_xi) cap U_nu[{y_zeta : zeta<xi}]. Nothing in the construction prevents this overlap set from being nonempty. That overlap lies in the finite levels and is not contained in U_r[{w^xi_n : n<omega}], so the proposed D_r(s,t)={w^xi_n : n<omega} does not satisfy Definition 2.3(b). Thus the amalgamation condition r is not shown to exist, and the omega_3-chain condition for P is not established. This is load-bearing for Theorem 5.1 and needs a different choice of D_r or of the adjoined points.
  2. [Lemma 4.10, Claim 4.10.2] The c.c.c. proof applies the stated Free Subset Theorem, which is formulated for functions f:omega_1 -> [omega_1]^omega, to sets E^i_delta and F^i_delta that are subsets of mu=omega_3. No reduction of omega_3 to omega_1 is given, and the proof does not state a generalized free-set theorem for subsets of omega_3. As written, the compatibility argument in Claim 4.10.2 is incomplete, and hence Theorem 4.9, which separates A-hat(omega_1,omega_1) from I-hat(omega_1,omega_1), is not established.
  3. [Theorem 4.4(1) and Section 4] The sufficient condition in Theorem 4.4(1) and the closing construction of Section 5 import from [8] the existence of LLSP spaces with constant cardinal sequence <omega_1>^alpha for every alpha<omega_3, and in Section 5 the existence of a space with cardinal sequence <omega_1>^{omega_2}. These are not proved or stated as exact theorems in the present paper. Since the new spaces are built by attaching copies of these spaces to new points, the new results inherit any flaw in those imported statements. Please state the exact imported theorems and give precise references or proof sketches.
minor comments (6)
  1. [Section 3 and references] The paper contains two missing bibliographic citations marked "[?]": the reduction theorem for C(alpha) in Section 3 and the Free Subset Theorem of Erdős and Specker in Lemma 4.10. These should be completed.
  2. [Theorem 3.3(2), proof] The proof contains a typo: "U(x) cap I_alpha(x)" should read "U(x) cap I_alpha(X)". In addition, the assertion that |U(x) cap I_alpha(X)| >= omega_1 is not immediate and needs a short justification.
  3. [Lemma 5.4, preparation] The text says "Fix an enumeration {y_xi : xi<omega_1} of Z_xi"; this should refer to Z_nu (or Z_nu \ X^*) rather than Z_xi.
  4. [Proposition 5.3, condition (C2)] Condition (C2) uses the symbol pi(t) without definition; it should be stated explicitly that pi(t) denotes the level of t.
  5. [Final paragraph of Section 5] The displayed cardinal sequence of the topological sum Z is stated as having lambda_alpha=omega for omega<alpha<omega_2, but since Y has cardinal sequence <omega_1>^{omega_2}, the sum has lambda_alpha=omega_1 on that interval. The subsequent argument does not seem to depend on this value, but the displayed sequence is incorrect.
  6. [Theorem 2.6, Claim 2.7.3] In the proof of Claim 2.7.3, the expression "U(p) \ U(q)" should be "U(p) \ V", since q has not been introduced at that point.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the derivation is self-contained given independent prior theorems.

full rationale

The paper's central constructions are not circular. Theorem 4.4 defines new cardinal functions A-hat and I-hat and proves bounds in terms of them; the sufficient direction builds an LLSP space from an almost disjoint family and previously constructed constant-sequence LLSP spaces, while the necessary direction derives I-hat from an arbitrary LLSP space. Neither direction fits a parameter to the conclusion. The main self-citations, to [8, Section 2] and [8, Theorem 2.1], supply black-box existence results for LLSP spaces with constant cardinal sequences; these are prior published theorems whose assumptions do not include the present target results, so they constitute independent support rather than circularity. Section 5's forcing construction is a genuine amalgamation argument using Proposition 5.3 and does not rename a fitted input as a prediction. The proof gap concerning Lemma 5.4's amalgamation, if real, is a correctness issue and not a reduction of the claim to its own inputs. Thus no derivation step is equivalent by construction to its assumptions.

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

The ledger contains six imported background and reference assumptions, no numerical free parameters, and no new axiomatic entities. The most notable debt is to the authors' prior paper [8], which supplies the constant-sequence LLSP spaces used as building blocks. The forcing posets and the cardinal functions  and Πare definitions, not postulates, so they are not listed as invented entities.

assumptions (6)
  • domain assumption For every ordinal α<ω3 there is an LLSP space with cardinal sequence <ω1>^α (the constant sequence of length α).
    Proved in the authors' earlier paper [8, Section 2], not reproved here; used in Lemma 4.6 and in the Section 4 construction to provide the building blocks Lσ. If false, the sufficient condition construction of Theorem 4.4(1) collapses.
  • domain assumption There exists an LLSP space Y with cardinal sequence <ω1>^ω2 ([8, Theorem 2.1]).
    Used in the final paragraph of Section 5 to build the sum space Z and argue that the reverse implication of Theorem 4.4(1) fails. Imported from the same authors' previous paper.
  • standard math Free Subset Theorem of Erdős-Specker: if f:ω1→[ω1]^ω and tp(f(α))<η for each α<ω1, then there is an uncountable f-free subset.
    Used in Claim 4.10.2 to prove that the forcing poset in Lemma 4.10 has the c.c.c.; the paper's text leaves the citation as '[?]' but the reference list includes [4].
  • standard math Erdős-Rado partition theorem: (2^ω)+ → (ω1)^2_ω, hence Â(ω1,ω1)=ω3 under GCH plus Cohen reals (Lemma 4.3).
    Used to show that the naive conjecture fails in the Cohen model after adding ω3 Cohen reals; from Baumgartner [2].
  • standard math Standard c.c.c. and ω2-closed forcing facts, including that c.c.c. extensions preserve cardinals and cofinalities.
    Used throughout Sections 4 and 5, for example in Theorem 4.8 and Proposition 5.3; standard forcing technology invoked without proof.
  • domain assumption Every LLSP space is 0-dimensional and admits an LLSP neighborhood assignment.
    Stated as Proposition 2.2 and attributed to [8, Proposition 1.1]; needed for the converse direction of the cone-system characterization in Theorem 2.6(2) and Corollary 2.7.

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Pith. "Pith review of Cardinal Sequences of Lindel\"of scattered P-spaces." pith.science (2026). https://pith.science/paper/WP2VSVXD

@misc{pith2026241117791,
  author       = {Pith},
  title        = {Pith review of: Cardinal Sequences of Lindel\"of scattered P-spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WP2VSVXD}},
  note         = {Machine review of arXiv:2411.17791}
}
abstract

We continue our investigation of cardinal sequences associated with locally Lindelof, scattered, Hausdorff P-spaces (abbreviated as LLSP spaces). We outline a method for constructing LLSP spaces from cone systems and partial orders with specific properties. Additionally, we establish limitations on the cardinal sequences of LLSP spaces. Finally, we present both a necessary condition and a distinct sufficient condition for a sequence $\langle \kappa_\alpha: \alpha < \omega_2 \rangle$ to be the cardinal sequence of an LLSP space.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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Reviewed August 12, 2026 · model on record in the stance chip above.