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The two-disjoint-copies property for compact spaces, homogeneity and connection with $C_p$-theory

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read No scattered compact space has the two-disjoint-copies property and ZFC yields perfect counterexamples.

desk verdict The paper cleanly separates 2DCP from scattered compacta in ZFC and gives explicit perfect counterexamples, which settles the metric case via Cantor-Bendixson. read the letter →

arxiv 2606.27568 v1 pith:TWN26MDY submitted 2026-06-25 math.GN

Jerzy Kąkol , Ondřej Kurka , Wies{l}aw Śliwa This is my paper · ORCID
classification math.GN
keywords two-disjoint-copiespropertycompactspacesscatteredperfectC_p-theoryEfimovproblemhomogeneouszero-dimensional
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 examines the two-disjoint-copies property on Tychonoff spaces, defined via a sequence of nonempty compact sets where each splits into two disjoint homeomorphic copies of the next. It proves that scattered compact spaces never satisfy this property. It also gives ZFC constructions of compact perfect spaces that fail the property. These facts together show that the property holds for a compact metric space exactly when the space is uncountable. The work further identifies sufficient conditions such as local homogeneity and supplies positive examples among certain zero-dimensional Efimov-type spaces while connecting the property to questions about metrizable quotients of C_p(X).

What carries the argument

The two-disjoint-copies property (2DCP), the existence of a sequence (K_n) of nonempty compact subsets of X such that each K_n contains two disjoint subsets homeomorphic to K_{n+1}.

What would settle it

An explicit construction of a scattered compact space that admits a sequence of nonempty compact subsets each containing two disjoint homeomorphic copies of the next one would falsify the claim that no scattered compact space has 2DCP.

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Extended reading notes

Core claim

No scattered compact space has 2DCP and there exist in ZFC compact perfect spaces without 2DCP. This implies that for compact metric spaces X the 2DCP is equivalent to uncountability of X. There exist explicit uncountable separable compact spaces failing 2DCP, for example the Isbell-Mrówka compacta. Positive classes among zero-dimensional compact spaces include the Brech as well as the Sobota-Zdomskyy compact spaces of Efimov type, which have 2DCP. Locally homogeneous compact spaces and spaces containing a copy of βω or 2^ω also have 2DCP, though compact h-homogeneous spaces with 2DCP that contain neither such copy exist in ZFC.

Load-bearing premise

The earlier result that 2DCP produces an infinite-dimensional metrizable quotient of C_p(X) continues to hold for the spaces considered here.

Editorial extensions

If this is right

  • Scattered compact spaces fail to have 2DCP.
  • Compact perfect spaces without 2DCP exist in ZFC.
  • For compact metric spaces, 2DCP holds precisely when the space is uncountable.
  • Locally homogeneous compact spaces have 2DCP.
  • Compact spaces containing a copy of βω or 2^ω have 2DCP.

Reading between the lines

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

  • Because 2DCP is sufficient but not necessary for all infinite compact spaces to produce the desired C_p(X) quotient, the open question whether every infinite compact X yields an infinite-dimensional metrizable quotient of C_p(X) remains unresolved by these examples.
  • The Isbell-Mrówka compacta supply concrete separable uncountable test spaces that can be checked directly for the presence or absence of infinite-dimensional metrizable quotients of their C_p spaces.
  • The existence of h-homogeneous compact spaces with 2DCP but without copies of βω or 2^ω shows that standard copies are sufficient but not required for the property.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper defines the two-disjoint-copies property (2DCP) for Tychonoff spaces: a space X has 2DCP if there is a sequence (K_n) of nonempty compact subsets such that each K_n contains two disjoint subsets homeomorphic to K_{n+1}. It proves that no scattered compact space has 2DCP, constructs explicit ZFC examples of perfect compacta without 2DCP (including Isbell-Mrówka compacta), shows that for compact metric spaces 2DCP is equivalent to uncountability, identifies classes that satisfy 2DCP (locally homogeneous spaces, spaces containing βω or 2^ω, Brech spaces, Sobota-Zdomskyy Efimov-type spaces) and classes that do not, and discusses the link to infinite-dimensional metrizable quotients of C_p(X) and Efimov's problem.

Significance. The results supply ZFC counterexamples to 2DCP among perfect compacta and a clean characterization for the metric case via the Cantor-Bendixson theorem. The explicit constructions (Isbell-Mrówka spaces) and positive results for specific zero-dimensional classes are concrete contributions that separate 2DCP from the mere presence of 2^ω copies and advance the motivating question about C_p(X) quotients.

minor comments (3)
  1. [Definition section] §1 (or wherever the definition of 2DCP is formalized): the recursive clause 'each K_n contains two disjoint subsets homeomorphic to K_{n+1}' should be stated with an explicit quantifier over the two subsets to avoid any ambiguity about whether the homeomorphisms are required to be onto the whole K_{n+1}.
  2. [Abstract / introduction] The statement that 'compact h-homogeneous spaces with 2DCP without such copies exist in ZFC' is asserted without a reference or section pointer in the abstract; ensure the construction is cross-referenced to the relevant theorem number in the body.
  3. [Examples section] Table or list of examples (if present): the Isbell-Mrówka compacta are cited as failing 2DCP; confirm that the argument uses only the standard properties of these spaces and does not rely on additional set-theoretic assumptions beyond ZFC.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, the clear summary of its contributions, and the recommendation to accept. We are pleased that the results on the two-disjoint-copies property, the ZFC examples, the metric characterization, and the connections to C_p-theory and Efimov's problem were viewed as concrete advances.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; central claims are independent new results

full rationale

The paper's core theorems (no scattered compact has 2DCP; explicit ZFC perfect compacta without 2DCP; equivalence for compact metric spaces via Cantor-Bendixson) rely on standard topological facts and new arguments, not on any self-citation or fitted input. The Banakh-Kąkol-Śliwa citation appears only in the abstract for motivational context about C_p(X) quotients and is explicitly not invoked in the new proofs. No equations, self-definitions, or load-bearing reductions exist. This is a normal non-circular case.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper works inside ZFC and standard topology; the abstract introduces the 2DCP definition but invokes no new free parameters or invented entities.

assumptions (1)
  • standard math ZFC set theory
    The paper states existence and non-existence results that hold in ZFC.

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Cite this review

Pith. "Pith review of The two-disjoint-copies property for compact spaces, homogeneity and connection with $C_p$-theory." pith.science (2026). https://pith.science/paper/TWN26MDY

@misc{pith2026260627568,
  author       = {Pith},
  title        = {Pith review of: The two-disjoint-copies property for compact spaces, homogeneity and connection with $C_p$-theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TWN26MDY}},
  note         = {Machine review of arXiv:2606.27568}
}
abstract

A Tychonoff space $X$ has the two-disjoint-copies property (2DCP) if there exists a sequence $(K_n)_{n\in\omega}$ of non-empty compact subsets of $X$ such that each $K_n$ contains two disjoint subsets homeomorphic to $K_{n+1}$. Banakh, K\k{a}kol and \'Sliwa showed that 2DCP yields an infinite-dimensional metrizable quotient of $C_p(X)$, while it is still a long-standing open question whether $C_p(X)$ has such a quotient for any infinite compact space $X$. The above concept as well as the last problem are closely related to Efimov's problem that has remained open for 40 years. We will discuss a number of conditions that imply 2DCP. For example, every locally homogeneous compact space, every space containing a copy of $\beta\omega$ or $2^\omega$ has 2DCP although compact $h$-homogeneous spaces with 2DCP without such copies exist in ZFC. We prove that no scattered compact space has 2DCP and there exist in ZFC compact perfect spaces without 2DCP. This implies that for compact metric spaces $X$ the 2DCP is equivalent to uncountability of $X$. There exist explicit uncountable separable compact spaces failing 2DCP, for example the Isbell-Mr\'owka compacta. We give positive classes among zero-dimensional compact spaces; for example, the Brech, as well as the Sobota-Zdomskyy compact spaces of Efimov type have 2DCP. Open questions are included.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Continuous Linear Surjections from $C_p(X)$ onto Symmetric Sequence Ideals in $c_0$

    math.FA 2026-08 accept novelty 7.0 of 10

    A non-zero symmetric sequence ideal E ⊆ c_0 is a continuous linear image of C_p(X) only if E = c_0, so proper ideals such as (ℓ_q)_p never appear.

  2. Talagrand compacta, 2DCP, and pointwise quotients

    math.GN 2026-07 conditional novelty 7.0 of 10

    Diamond-guided Talagrand compacta can fail 2DCP and local homogeneity while remaining Grothendieck Efimov spaces, and no Talagrand compactum admits classical pointwise sequence quotients.

Reference graph

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