REVIEW 3 major objections 5 minor 10 references
Zero-dimensional metrizable CDH space $X$ such that $X^2$ is not CDH
T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read This paper constructs, in ZFC alone, a zero-dimensional metrizable CDH space $X$ whose square $X^2$ is not CDH, settling an open question.
desk verdict Plausible ZFC construction for an open CDH problem, but the submitted proof has a fatal contradiction in the counting argument. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the disjoint sum $X=Y\oplus 2^\omega$ with $Y$ a CDH $\lambda$-set; a $\lambda$-set is a subspace of $2^\omega$ in which every countable subset is relatively $G_\delta$, and no $\lambda$-set contains a copy of $2^\omega$. The invariance step is carried by the paper's Corollary 5, derived from the proof of a known product theorem: in a product of a space that contains $2^\omega$ with one that does not, some countable dense set has no countable subset homeomorphic to $2^\omega$. This corollary certifies that the component $Q_0$ of the constructed dense set cannot absorb a Cantor set, while Lemma 3 supplies $\mathfrak{c}$ pairwise nonhomeomorphic nowhere dense subsets of $\mathbb{Q}$ used as tags $C_r$.
What would settle it
Check whether $Y\times Y$ contains a copy of $2^\omega$ for the $Y$ supplied by Theorem 2; if it does, the paper's invariance argument fails. Equally, if some clopen subset of $Y$ is not crowded, the projection step no longer yields a copy of $2^\omega$, and the conclusion that $h$ preserves $2^\omega\times Y$ has no basis.
Extended reading notes
Core claim
Set $X=Y\oplus 2^\omega$, where $Y$ is a CDH $\lambda$-set of cardinality $\aleph_1$; Theorem 2 supplies such a $Y$ for any cardinal up to $\mathfrak{b}$. The paper proves that every homeomorphism of $X^2$ must preserve the clopen summand $2^\omega\times Y$: a clopen piece of that summand contains a copy of $2^\omega$, while $Y^2$ does not, and the projection of any Cantor set that tried to cross into $2^\omega\times Y$ would give a clopen crowded subset of $Y$, hence a copy of $2^\omega$ inside $Y$. Inside $2^\omega\times Y$, the proof builds $\mathfrak{c}$ countable dense subsets $D_r$ by taking one fixed dense set $D=Q_0\cup Q_1$ and adjoining pairwise nonhomeomorphic nowhere dense sets $C_r\subseteq\{s\}\times Y$. Any homeomorphism moving $D_p$ to $D_r$ would have to send a point of $\{s\}\times Y$ into a Cantor-set half, forcing a copy of $2^\omega$ into the $\lambda$-set side; the resulting contradiction shows the types are distinct, so $X^2$ is not CDH.
Load-bearing premise
The proof needs the unproved fact that $Y^2$ contains no copy of $2^\omega$ (and that every clopen subset of $Y$ is crowded); if a CDH $\lambda$-set of cardinality $\aleph_1$ failed either condition, the step forcing every homeomorphism to preserve $2^\omega\times Y$ would collapse.
Editorial extensions
If this is right
- The square of the constructed space has exactly $\mathfrak{c}$ types of countable dense subsets, so the open question is answered negatively: CDH is not preserved by squares even in ZFC.
- The construction also settles the $\kappa=\mathfrak{c}$ case of the question asking which cardinals can occur as the number of types for such a space.
- The existence of the space is unconditional, unlike the earlier example that used an additional set-theoretic axiom.
- Consistently, the example is a co-analytic subspace of $2^\omega$, which is best possible in the sense that Borel zero-dimensional CDH spaces are classified and have CDH squares.
Reading between the lines
- A natural extension, not stated in the paper, is that the same construction may make $X^n$ non-CDH for every $n\ge 2$ by applying the clopen-summand argument to $2^\omega\times Y^{n-1}$; the needed check is whether $Y^k$ remains free of copies of $2^\omega$ for all $k$.
- The counting mechanism is modular: it only needs one factor with a Cantor-set neighborhood and one Cantor-free factor with enough nonhomeomorphic nowhere dense subsets, so the 'exactly $\mathfrak{c}$ types' conclusion may hold for a broader class of CDH spaces than the particular sum $Y\oplus 2^\omega$.
- Because the construction is in ZFC while the descriptive upgrade is only consistent, the boundary between ZFC examples and projective examples appears to be genuinely set-theoretic; this is suggested by the paper's reliance on a consistency result for co-analyticity.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a ZFC construction of a zero-dimensional metrizable CDH space X with the property that X^2 has exactly c types of countable dense subsets, and consistently this space can be co-analytic. The construction takes a CDH λ-set Y of cardinality aleph_1, forms X = Y ⊕ 2^ω, and attempts to show that every homeomorphism of X^2 preserves the clopen summand 2^ω × Y, after which a family {D_r} of countable dense subsets of 2^ω × Y indexed by r ∈ (0,1) is built by adjoining pairwise nonhomeomorphic countable nowhere dense sets C_r to a common countable core D. The paper claims to answer an open question of Medini and to give a consistent co-analytic example. The proof is forward and uses external theorems ([1], [3], [4], [10]) in a non-circular way.
Significance. If the main theorem were established, it would answer a question of Medini and complement the earlier consistent construction under MA(σ-centered) with a ZFC example. The use of λ-sets and the strategy of identifying a clopen summand preserved by all homeomorphisms are natural and potentially useful. The proof is not circular: it relies on previously known results rather than on its own conclusions. However, the central counting argument contains an impossible object, and several supporting claims about the λ-set Y are not justified. As written, the main theorem is not established.
major comments (3)
- [§3, Theorem 7] The definition of F_n is internally inconsistent: the text says 'For n ∈ ω let F_n ⊂ U_n be countable such that F_n ≈ 2^ω.' No countable set is homeomorphic to the Cantor set. If 'countable' is taken literally, then h(F_n) is countable for every homeomorphism h, so it cannot contradict Corollary 5's statement 'for no D ⊂ C we have D ≈ 2^ω'; the alleged contradiction disappears. If '≈ 2^ω' is taken literally, then Q1 = ⋃ F_n is uncountable, so D = Q0 ∪ Q1 is not a countable dense subset, and the construction does not produce countable dense sets D_r of the desired kind. This is not a missing justification but an impossible requirement at the center of the c-types counting argument.
- [§3, Theorem 7] The step 'By Lemma 3, there is a collection {C_r; r ∈ (0,1)} of countable pairwise nonhomeomorphic nowhere dense subsets of {s} × Y' is not justified. Lemma 3 gives pairwise nonhomeomorphic nowhere dense subsets of Q, but the paper does not prove that these spaces can be realized as countable nowhere dense subsets of {s} × Y. This would require, for instance, an embedding of Q (or of each relevant countable space) into {s} × Y and a verification that the embedded copies are nowhere dense in {s} × Y. Without this transfer, the final contradiction h(C_p) = C_r is unsupported.
- [§3, Theorem 7] The claim that π2(h(2^ω) ∩ (2^ω × Y)) 'is a clopen subset of Y therefore it is a crowded space' assumes without proof that Y has no isolated points. A CDH λ-set can have isolated points (for example, a space homeomorphic to 2^ω with one isolated point added is CDH), and a clopen subset of such a space need not be crowded. Since the conclusion that this set is homeomorphic to 2^ω is used to prove h(2^ω) = 2^ω, this missing hypothesis is load-bearing for the invariance argument.
minor comments (5)
- [Abstract] The abstract states that X^2 'has exactly c countable dense subsets', whereas the body and the intended theorem concern 'exactly c many types of countable dense subsets'. These are different statements and the abstract should be corrected.
- [§2, Remark after Definition 1] The paper repeatedly uses the fact that Y^2 contains no copy of 2^ω, but only the remark that no λ-set contains a copy of 2^ω is stated. The product statement is plausibly true for λ-sets, but it should be proved or explicitly cited.
- [§3, Theorem 7] The phrase 'any clopen subset of 2^ω × Y contains the Cantor set' should be qualified to nonempty clopen subsets; otherwise the statement is false for the empty set.
- [§3, Theorem 7] The decomposition 'X^2 ≈ Y^2 ⊕ 2^ω ⊕ 2^ω × Y' compresses the full expansion (Y ⊕ 2^ω)^2; it would help to note explicitly that the two cross-product summands are identified under 2^ω ⊕ 2^ω ≈ 2^ω and that 2^ω × 2^ω ≈ 2^ω.
- [§3, Theorem 6] In the statement of Theorem 6, the notation 'ω L 1' is garbled; it should presumably read '(ω_1)^L'.
Circularity Check
No circularity found; the construction is a forward argument from external theorems, although Theorem 7 contains a separate internal inconsistency.
full rationale
The derivation chain is not circular. The space Y is obtained from Theorem 2 of reference [4], an external prior result, and X = Y ⊕ 2^ω is then assembled; the CDH property of X is inherited from the CDH property of the two summands, with no fitted parameters and no appeal to the target conclusion. The invariance of the clopen summand 2^ω × Y under homeomorphisms of X^2 is argued from the external λ-set fact that Y contains no copy of 2^ω and from standard Cantor-set characterizations, not from the conclusion being proved. The counting of types uses Lemma 3 of reference [1] for nonhomeomorphic nowhere dense subsets and a cardinality bound. There are no self-citations and no parameter is fitted to the claimed result. A genuine non-circular difficulty exists in the proof of Theorem 7: the line 'For n ∈ ω let F_n ⊂ U_n be countable such that F_n ≈ 2^ω' requires a countable space homeomorphic to the Cantor set, which is impossible, and Corollary 5's property 'for no countable subset E ⊂ Q0 we have E ≈ 2^ω' is vacuously true of every countable Q0. These defects undermine the counting argument as a matter of mathematical correctness, but they are not circularity: the conclusion is not equivalent to the inputs by construction, and no load-bearing self-citation chain is present.
Assumptions & free parameters
assumptions (6)
- domain assumption There exists a CDH lambda-set Y of cardinality aleph_1 (Theorem 2 of [4]).
- domain assumption Y^2 contains no copy of 2^omega.
- domain assumption Every clopen subset of Y is crowded (Y has no isolated points).
- domain assumption Corollary 5 of [3]: there is a countable dense C in X times Y such that no subset has closure homeomorphic to 2^omega.
- domain assumption Theorem 6 of [10]: under MA + not CH + omega_1 = (omega_1)^L, every subset of 2^omega of cardinality aleph_1 is Pi^1_1.
- domain assumption Lemma 3 of [1]: there exist c many pairwise nonhomeomorphic nowhere dense subsets of Q.
Cite this review
Pith. "Pith review of Zero-dimensional metrizable CDH space $X$ such that $X^2$ is not CDH." pith.science (2026). https://pith.science/paper/6LNH7VDD
@misc{pith2026241117573,
author = {Pith},
title = {Pith review of: Zero-dimensional metrizable CDH space $X$ such that $X^2$ is not CDH},
year = {2026},
howpublished = {\url{https://pith.science/paper/6LNH7VDD}},
note = {Machine review of arXiv:2411.17573}
}
abstract
In this paper a construction of a metrizable zero-dimensional CDH space $X$ such that $X^2$ has exactly $\mathfrak{c}$ countable dense subsets is provided. Furthermore, it is shown that the space can be constructed consistently co-analytic. Thus answering an open question asked by Medini. To do so we use the notion of $\lambda$-sets.
Reference graph
Works this paper leans on
- [1]
-
[3]
R. Hern´ andez-Guti´ errez. Countable dense homogeneit y and the double arrow space. Topology Appl., 160(10):1123–1128, 2013
work page 2013
-
[4]
R. Hern´ andez-Guti´ errez, M. Hruˇ s´ ak, and J. van Mill.Countable dense homogeneity and λ-sets. Fund. Math., 226(2):157–172, 2014
work page 2014
-
[10]
A. W. Miller. Descriptive set theory and forcing , volume 4 of Lecture Notes in Logic . Springer-Verlag, Berlin, 1995. How to prove theorems about Borel sets the hard way. 5
work page 1995
-
[2]
B. Fitzpatrick, Jr. and H. X. Zhou. Countable dense homog eneity and the Baire property. Topology Appl., 43(1):1–14, 1992
work page 1992
-
[5]
M. Hruˇ s´ ak and J. van Mill. Open problems on countable de nse homogeneity. Topology Appl., 241:185–196, 2018
work page 2018
-
[6]
M. Hruˇ s´ ak and B. Zamora Avil´ es. Countable dense homog eneity of definable spaces. Proc. Amer. Math. Soc. , 133(11):3429–3435, 2005
work page 2005
-
[7]
M. Hruˇ s´ ak and J. van Mill. Nearly countable dense homog eneous spaces. Canad. J. Math., 66(4):743–758, 2014
work page 2014
Show all 10 references
-
[8]
T. Jech. Set theory . Springer Monographs in Mathematics. Springer-Verlag, Be rlin, millennium edition, 2003
2003
-
[9]
A. Medini. Products and countable dense homogeneity. Topology Proc., 46:135–143, 2015. 4
2015
Reviewed August 12, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.