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arxiv: 2606.21204 · v1 · pith:KWLGMCRLnew · submitted 2026-06-19 · 🧮 math.FA

Borel complexity of isometry classes of mathcal{C}(K) spaces with countable compacta

Marek C\'uth , Martin Dole\v{z}al , Ond\v{r}ej Kurka , Jakub Rondo\v{s} This is my paper

Pith reviewed 2026-06-26 13:18 UTC · model grok-4.3

classification 🧮 math.FA
keywords Borel complexityisometry classesC(K) spacescountable compactaBanach space classificationdescriptive set theoryL1-preduals
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The pith

The isometry class of C(K) for each countable compact K has exactly determined Borel complexity.

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

The paper shows that for any countable compact space K the set of separable Banach spaces isometric to C(K) sits at a precise level in the Borel hierarchy. This level is fixed by the topological properties of K. A sympathetic reader cares because the result supplies concrete natural examples of sets whose Borel complexity is both arbitrarily high and fully specified, rather than merely existent at some level. The work also determines the exact Borel complexity of homeomorphism classes of countable compacta and gives a characterization of those real L1-preduals that arise as C(K) for zero-dimensional compact K.

Core claim

For every countable compact space K the isometry class of the Banach space C(K) is exactly classifiable at a Borel complexity level determined by the properties of K inside the standard Borel structure on the space of separable Banach spaces.

What carries the argument

The isometry relation on the space of separable Banach spaces equipped with its standard Borel structure, with each C(K) class whose complexity tracks the topology of the countable compactum K.

If this is right

  • The homeomorphism class of every countable compact K has its Borel complexity exactly determined.
  • Natural examples exist of sets at arbitrarily high yet precisely known levels of the Borel hierarchy.
  • Real L1-preduals isometric to C(K) for zero-dimensional K receive a new characterization.
  • The isometry class of C(2^N) receives an exact Borel complexity assignment.

Where Pith is reading between the lines

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

  • The same descriptive-set-theoretic techniques might resolve isometry classification for other families of Banach spaces.
  • Classification questions in functional analysis can be settled with sharp rather than merely existential bounds.
  • The results open the possibility of effective or computable versions of these classifications.

Load-bearing premise

The isometry relation on separable Banach spaces admits a standard Borel structure in which each C(K) class is a Borel set whose complexity is fixed by the topology of K.

What would settle it

Exhibiting one countable compact K for which the isometry class of C(K) lies strictly higher in the Borel hierarchy than the level predicted from the topological invariants of K.

read the original abstract

For every countable compact space $K$, we determine the exact Borel complexity of the isometry class of the Banach space $\mathcal{C}(K)$. As a byproduct, we also determine the precise Borel complexity of the homeomorphism class of a fixed countable compact space $K$, improving earlier results of Cenzer and Mauldin. The above results provide concrete and natural examples of sets with arbitrarily high, still exactly determined, Borel complexity. Moreover, we find a new characterization of those real $L_1$-preduals that are isometric to $\mathcal{C}(K)$ for some zero-dimensional compact space $K$ and we determine the precise Borel complexity of $\mathcal{C}(2^{\mathbb{N}})$.

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

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Summary. The manuscript determines the exact Borel complexity of the isometry class of the Banach space C(K) for every countable compact space K. As a byproduct, it determines the precise Borel complexity of the homeomorphism class of each such K (improving Cenzer-Mauldin), provides a new characterization of those real L1-preduals isometric to C(K) for some zero-dimensional compact K, and computes the Borel complexity of C(2^N). The results are obtained via the Banach-Stone theorem, Cantor-Bendixson analysis of K, and explicit reductions in the standard Effros Borel structure on closed subspaces of C[0,1], yielding concrete examples of sets with arbitrarily high but exactly determined Borel complexity.

Significance. If the results hold, the paper supplies natural examples of sets with precisely determined Borel complexities at arbitrary levels of the hierarchy, strengthening the interface between descriptive set theory and Banach space theory. The explicit tracking of Borel classes through derivatives and the parameter-free derivations from standard codings of separable Banach spaces and compacta are methodological strengths that yield falsifiable predictions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation to accept. We are pleased that the results are recognized as providing natural examples of sets with precisely determined Borel complexities and as strengthening connections between descriptive set theory and Banach space theory.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes exact Borel complexities for isometry classes of C(K) via the Banach-Stone theorem, an explicit new characterization of relevant L1-preduals, and Cantor-Bendixson derivatives on countable compacta K. Upper and lower bounds are obtained through direct reductions whose Borel classes are tracked explicitly in the Effros Borel structure on subspaces of C[0,1]. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear; the cited prior result of Cenzer-Mauldin is external and the improvement follows from independent invariants. The derivation is self-contained against standard descriptive-set-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5665 in / 1103 out tokens · 20880 ms · 2026-06-26T13:18:51.260341+00:00 · methodology

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Works this paper leans on

48 extracted references · 29 canonical work pages

  1. [1]

    Polish spaces of

    C. Polish spaces of. Forum Math. Sigma , FJOURNAL =. 2022 , PAGES =. doi:10.1017/fms.2022.16 , URL =

    work page doi:10.1017/fms.2022.16 2022

  2. [2]

    Polish spaces of

    C. Polish spaces of. J. Inst. Math. Jussieu , volume =. 2024 , number =

    2024

  3. [3]

    Daniel , TITLE =

    Cenzer, Douglas and Mauldin, R. Daniel , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 1983 , NUMBER =

    1983

  4. [4]

    Notre Dame J

    Westrick, Linda , TITLE =. Notre Dame J. Form. Log. , FJOURNAL =. 2020 , NUMBER =. doi:10.1215/00294527-2020-0006 , URL =

    work page doi:10.1215/00294527-2020-0006 2020

  5. [5]

    and Woodin, W

    Kechris, Alexander S. and Woodin, W. Hugh , TITLE =. Mathematika , FJOURNAL =. 1986 , NUMBER =. doi:10.1112/S0025579300011244 , URL =

    work page doi:10.1112/s0025579300011244 1986

  6. [6]

    Westrick, Linda Brown , TITLE =. J. Symb. Log. , FJOURNAL =. 2014 , NUMBER =. doi:10.1017/jsl.2013.23 , URL =

    work page doi:10.1017/jsl.2013.23 2014

  7. [7]

    Daniel , TITLE =

    Cenzer, Douglas and Mauldin, R. Daniel , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 1982 , NUMBER =

    1982

  8. [8]

    , TITLE =

    Kechris, Alexander S. , TITLE =. 1995 , PAGES =

    1995

  9. [9]

    Lancien, Gilles , TITLE =. RACSAM. Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. , FJOURNAL =. 2006 , NUMBER =

    2006

  10. [10]

    Casini, Emanuele and Miglierina, Enrico and Piasecki, Lukasz , TITLE =. Topol. Methods Nonlinear Anal. , FJOURNAL =. 2024 , NUMBER =. doi:10.12775/tmna.2023.009 , URL =

    work page doi:10.12775/tmna.2023.009 2024

  11. [11]

    , TITLE =

    Alspach, Dale E. , TITLE =. Banach spaces (. 1993 , ISBN =. doi:10.1090/conm/144/1209442 , URL =

    work page doi:10.1090/conm/144/1209442 1993

  12. [12]

    Casini, Emanuele and Miglierina, Enrico and Piasecki, ukasz , TITLE =. Canad. Math. Bull. , FJOURNAL =. 2015 , NUMBER =. doi:10.4153/CMB-2015-024-9 , URL =

    work page doi:10.4153/cmb-2015-024-9 2015

  13. [13]

    monograph, Preliminary version available at

    Kadets, Vladimir and Mart\'in, Miguel and Rueda Zoca, Abraham and Werner, Dirk , TITLE =. monograph, Preliminary version available at

  14. [14]

    Elton , TITLE =

    Lacey, H. Elton , TITLE =. 1974 , PAGES =

    1974

  15. [15]

    Biorthogonal systems in

    H\'. Biorthogonal systems in. 2008 , PAGES =

    2008

  16. [16]

    Lazar, A. J. , TITLE =. Duke Math. J. , FJOURNAL =. 1972 , PAGES =

    1972

  17. [17]

    Rondo. The. J. Funct. Anal. , FJOURNAL =. 2021 , NUMBER =. doi:10.1016/j.jfa.2021.109251 , URL =

    work page doi:10.1016/j.jfa.2021.109251 2021

  18. [18]

    Integral representation theory , SERIES =

    Luke. Integral representation theory , SERIES =. 2010 , PAGES =

    2010

  19. [19]

    Baire classes of affine vector-valued functions , JOURNAL =

    Kalenda, Ond. Baire classes of affine vector-valued functions , JOURNAL =. 2016 , NUMBER =. doi:10.4064/sm8278-5-2016 , URL =

    work page doi:10.4064/sm8278-5-2016 2016

  20. [20]

    L\'opez-P\'erez, Gin\'es and Mart\'inez Va\ n\'o, Esteban and Rueda Zoca, Abraham , TITLE =. Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSAM , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s13398-025-01809-x , URL =

    work page doi:10.1007/s13398-025-01809-x 2026

  21. [21]

    2009 , PAGES =

    Givant, Steven and Halmos, Paul , TITLE =. 2009 , PAGES =. doi:10.1007/978-0-387-68436-9 , URL =

    work page doi:10.1007/978-0-387-68436-9 2009

  22. [22]

    1989 , PAGES =

    Koppelberg, Sabine , TITLE =. 1989 , PAGES =

    1989

  23. [23]

    and Marcone, Alberto , TITLE =

    Camerlo, Riccardo and Darji, Udayan B. and Marcone, Alberto , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2005 , NUMBER =. doi:10.1090/S0002-9947-05-03956-5 , URL =

    work page doi:10.1090/s0002-9947-05-03956-5 2005

  24. [24]

    and Kalton, N

    Godefroy, G. and Kalton, N. J. and Lancien, G. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2001 , NUMBER =. doi:10.1090/S0002-9947-01-02825-2 , URL =

    work page doi:10.1090/s0002-9947-01-02825-2 2001

  25. [25]

    Godefroy, Gilles and Saint-Raymond, Jean , TITLE =. J. Funct. Anal. , FJOURNAL =. 2018 , NUMBER =. doi:10.1016/j.jfa.2018.01.018 , URL =

    work page doi:10.1016/j.jfa.2018.01.018 2018

  26. [26]

    MLQ Math

    Melleray, Julien , TITLE =. MLQ Math. Log. Q. , FJOURNAL =. 2007 , NUMBER =. doi:10.1002/malq.200610032 , URL =

    work page doi:10.1002/malq.200610032 2007

  27. [27]

    Israel J

    Melleray, Julien , TITLE =. Israel J. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s11856-020-1976-1 , URL =

    work page doi:10.1007/s11856-020-1976-1 2020

  28. [28]

    , TITLE =

    Bourgain, J. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1980 , NUMBER =. doi:10.2307/2043243 , URL =

    work page doi:10.2307/2043243 1980

  29. [29]

    and Dodos, Pandelis , TITLE =

    Argyros, Spiros A. and Dodos, Pandelis , TITLE =. Adv. Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.aim.2006.05.013 , URL =

    work page doi:10.1016/j.aim.2006.05.013 2007

  30. [30]

    and Godefroy, Gilles and Rosenthal, Haskell P

    Argyros, Spiros A. and Godefroy, Gilles and Rosenthal, Haskell P. , TITLE =. Handbook of the geometry of. 2003 , ISBN =. doi:10.1016/S1874-5849(03)80030-X , URL =

    work page doi:10.1016/s1874-5849(03)80030-x 2003

  31. [31]

    and Beanland, K

    Antunes, L. and Beanland, K. and Braga, B. M. , TITLE =. Forum Math. Sigma , FJOURNAL =. 2021 , PAGES =. doi:10.1017/fms.2020.68 , URL =

    work page doi:10.1017/fms.2020.68 2021

  32. [32]

    , TITLE =

    Cuellar Carrera, W. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/tran/7319 , URL =

    work page doi:10.1090/tran/7319 2018

  33. [33]

    Bossard, Beno\^it , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 1993 , NUMBER =

    1993

  34. [34]

    Bossard, Beno\^it , TITLE =. Fund. Math. , FJOURNAL =. 2002 , NUMBER =. doi:10.4064/fm172-2-3 , URL =

    work page doi:10.4064/fm172-2-3 2002

  35. [35]

    Szlenk and w

    Ryan Michael Causey , keywords =. Szlenk and w. J. Math. Anal. Appl. , volume =. 2017 , issn =. doi:https://doi.org/10.1016/j.jmaa.2016.10.031 , url =

    work page doi:10.1016/j.jmaa.2016.10.031 2017

  36. [36]

    Ryan Michael Causey , journal=. The. 2016 , volume=

    2016

  37. [37]

    2009 , PAGES =

    Gao, Su , TITLE =. 2009 , PAGES =

    2009

  38. [38]

    Cuellar Carrera, Wilson and de Rancourt, No\'e. Local. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2023 , NUMBER =. doi:10.4171/jems/1257 , URL =

    work page doi:10.4171/jems/1257 2023

  39. [39]

    Godefroy, Gilles , TITLE =. J. Nonlinear Convex Anal. , FJOURNAL =. 2017 , NUMBER =

    2017

  40. [40]

    The isomorphism class of

    Kurka, Ond. The isomorphism class of. Israel J. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s11856-019-1851-0 , URL =

    work page doi:10.1007/s11856-019-1851-0 2019

  41. [41]

    , TITLE =

    Smith, Richard J. , TITLE =. 2026 , NOTE =

    2026

  42. [42]

    2026 , NOTE =

    C\'uth, Marek and de Rancourt, No\'e. 2026 , NOTE =

    2026

  43. [43]

    , TITLE =

    Lindenstrauss, Joram and Wulbert, Daniel E. , TITLE =. J. Functional Analysis , FJOURNAL =. 1969 , PAGES =. doi:10.1016/0022-1236(69)90003-2 , URL =

    work page doi:10.1016/0022-1236(69)90003-2 1969

  44. [44]

    Lindenstrauss, Joram , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 1964 , PAGES =

    1964

  45. [45]

    H\'ajek, Petr and Lancien, Gilles and Proch\'azka, Anton\'in , TITLE =. J. Math. Anal. Appl. , FJOURNAL =. 2009 , NUMBER =. doi:10.1016/j.jmaa.2008.11.075 , URL =

    work page doi:10.1016/j.jmaa.2008.11.075 2009

  46. [46]

    2011 , PAGES =

    Fabian, Mari\'an and Habala, Petr and H\'ajek, Petr and Montesinos, Vicente and Zizler, V\'aclav , TITLE =. 2011 , PAGES =. doi:10.1007/978-1-4419-7515-7 , URL =

    work page doi:10.1007/978-1-4419-7515-7 2011

  47. [47]

    Topology Appl

    Debs, Gabriel and Saint Raymond, Jean , TITLE =. Topology Appl. , FJOURNAL =. 2018 , PAGES =. doi:10.1016/j.topol.2018.07.010 , URL =

    work page doi:10.1016/j.topol.2018.07.010 2018

  48. [48]

    Miery slabej nekompaktnosti v klasick\'ych priestoroch , JOURNAL =

    Smer. Miery slabej nekompaktnosti v klasick\'ych priestoroch , JOURNAL =