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arxiv: 2512.00415 · v2 · submitted 2025-11-29 · 🧮 math.GN

Resolvability in products of spaces of small cardinality

Anton Lipin This is my paper

Pith reviewed 2026-05-17 03:49 UTC · model grok-4.3

classification 🧮 math.GN
keywords resolvabilityisodyne spacesproduct spacesregular spacesHausdorff spacescardinal ω₁general topology
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The pith

The product of any two regular isodyne spaces of cardinality ω₁ is ω-resolvable.

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

The paper proves that the product of any two regular isodyne spaces of cardinality ω₁ is ω-resolvable. It also proves that the product of any n+2 Hausdorff isodyne spaces of cardinality ω_n is ω-resolvable. Isodyne spaces are those in which every nonempty open set has the same cardinality as the whole space. These results matter because they identify conditions under which products of topological spaces admit a decomposition into countably many pairwise disjoint dense subsets. The proofs use the uniformity of open-set cardinalities together with the given separation axioms to build the required dense sets in the product.

Core claim

We prove that the product of any two regular isodyne spaces of cardinality ω₁ is ω-resolvable, and that the product of any n + 2 Hausdorff isodyne spaces of cardinality ω_n is ω-resolvable.

What carries the argument

The isodyne property (every nonempty open set has cardinality equal to the space) combined with regularity or the Hausdorff axiom at these small cardinals, which permits explicit construction of ω many disjoint dense subsets in the product.

If this is right

  • Such a product space can be written as the union of ω many pairwise disjoint dense subsets.
  • The property holds for arbitrary choices of the factor spaces as long as each satisfies the stated hypotheses.
  • The number of factors needed to guarantee ω-resolvability grows with the index n of the cardinal ω_n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar resolvability conclusions might hold for products involving more factors at larger cardinals if the isodyne condition is retained.
  • The result raises the question of whether the isodyne assumption can be weakened while keeping the product ω-resolvable.
  • One could look for a counterexample at ω₁ with only the Hausdorff axiom instead of regularity.

Load-bearing premise

The spaces are isodyne, so every nonempty open set has the same cardinality as the space, and they satisfy regularity or the Hausdorff separation axiom.

What would settle it

Two regular isodyne spaces of cardinality ω₁ whose product cannot be partitioned into countably many pairwise disjoint dense subsets would falsify the first claim.

Figures

Figures reproduced from arXiv: 2512.00415 by Anton Lipin.

Figure 1
Figure 1. Figure 1: A rough visualization of the defined sets [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

We prove that: I. The product of any two regular isodyne spaces of cardinality $\omega_1$ is $\omega$-resolvable; II. The product of any $n + 2$ Hausdorff isodyne spaces of cardinality $\omega_n$ is $\omega$-resolvable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves two main results on ω-resolvability of products: (I) the product of any two regular isodyne spaces of cardinality ω₁ is ω-resolvable, and (II) the product of any n+2 Hausdorff isodyne spaces of cardinality ω_n is ω-resolvable. The proofs rely on the isodyne property (every nonempty open set has full cardinality) together with regularity or the Hausdorff axiom to construct countably many pairwise disjoint dense subsets via iterative selection from open sets in the product topology.

Significance. If the results hold, they provide concrete positive instances of resolvability in products of small-cardinality spaces under standard separation and uniformity conditions. The explicit iterative constructions and the use of pigeonhole arguments on coordinates for the Hausdorff case constitute a clear technical contribution to the literature on resolvability and isodyne spaces. The absence of free parameters or ad-hoc axioms in the derivations is a strength.

minor comments (3)
  1. §2, Definition of isodyne: the parenthetical remark on open sets of full cardinality could be expanded with a brief comparison to the standard definition in the literature to aid readers unfamiliar with the term.
  2. Proof of Theorem 3.2 (part I): the induction step on the countable collection of dense sets would benefit from an explicit statement of how regularity ensures that the closure of the chosen point does not intersect previously selected sets in the product.
  3. §4, final paragraph: the remark on extending the result beyond ω_n appears informal; either remove it or supply a precise conjecture with supporting heuristic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive report, including the accurate summary of our two main theorems and the recognition of the technical contribution from the iterative constructions and pigeonhole arguments. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper supplies direct, self-contained proofs of two new resolvability statements for products of isodyne spaces. The arguments proceed by explicit iterative selection of points from nonempty open sets whose cardinalities remain equal to the full space cardinality under the isodyne hypothesis, using regularity only to control closures in the product topology and a pigeonhole argument on coordinates for the Hausdorff case. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the theorems are established from the stated separation and uniformity assumptions without circular reduction to their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions of isodyne spaces, resolvability, regularity, and Hausdorff separation from prior literature, together with the axioms of ZFC for handling cardinals ω_n. No free parameters, new entities, or ad-hoc assumptions are introduced in the abstract.

axioms (2)
  • standard math ZFC set theory for cardinal arithmetic and existence of spaces of cardinality ω_n
    Invoked implicitly to construct or reason about spaces of the stated cardinalities and their products.
  • domain assumption Definitions of isodyne space and ω-resolvability from prior topological literature
    The theorems are stated in terms of these established notions; the abstract does not redefine them.

pith-pipeline@v0.9.0 · 5323 in / 1389 out tokens · 105949 ms · 2026-05-17T03:49:21.555076+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

  1. [1]

    Bešlagić, R

    A. Bešlagić, R. Levy, Irresolvable products,Papers on general topology and applications (Gorham, ME, 1995), Ann. New York Acad. Sci.,806 (1996), 42—48

    work page 1995

  2. [2]

    Bhaskara Rao A note on resolvability,Acta Mathematica Hun- garica,159(2019), 669-673

    K.P.S. Bhaskara Rao A note on resolvability,Acta Mathematica Hun- garica,159(2019), 669-673

    work page 2019

  3. [3]

    Ceder, On maximally resolvable spaces,Fundamenta Mathemati- cae,55(1964), 87–93

    J.G. Ceder, On maximally resolvable spaces,Fundamenta Mathemati- cae,55(1964), 87–93

    work page 1964

  4. [4]

    Illanes, Finite and𝜔-resolvability,Proceedings of the American math- ematical society,124:4 (April 1996), 1243-1246

    A. Illanes, Finite and𝜔-resolvability,Proceedings of the American math- ematical society,124:4 (April 1996), 1243-1246. 9

    work page 1996

  5. [5]

    Juhász, L

    I. Juhász, L. Soukup, Z. Szentmiklóssy, On resolvability of products, Fundamenta Mathematicae260(2023), 281–295

    work page 2023

  6. [6]

    Lipin, Resolvability in products and squares,Acta Mathematica Hungarica(2025) DOI: 10.1007/s10474-025-01565-9

    A.E. Lipin, Resolvability in products and squares,Acta Mathematica Hungarica(2025) DOI: 10.1007/s10474-025-01565-9

    work page doi:10.1007/s10474-025-01565-9 2025

  7. [7]

    Pavlov, On resolvability of topological spaces,Topology and its Ap- plications,126:1-2 (2002), 37-47

    O. Pavlov, On resolvability of topological spaces,Topology and its Ap- plications,126:1-2 (2002), 37-47

    work page 2002

  8. [8]

    Pavlov, Problems on (ir)resolvability,Open Problems in Topology II (2007), Elsevier B.V

    O. Pavlov, Problems on (ir)resolvability,Open Problems in Topology II (2007), Elsevier B.V. 10

    work page 2007