Lotz-Peck-Porta and Rosenthal's theorems for spaces $C_p(X)$

Jerzy K\k{a}kol; Ond\v{r}ej Kurka; Wies{\l}aw \'Sliwa

arxiv: 2509.08634 · v1 · submitted 2025-09-10 · 🧮 math.FA · math.GN

Lotz-Peck-Porta and Rosenthal's theorems for spaces C_p(X)

Jerzy K\k{a}kol , Ond\v{r}ej Kurka , Wies{\l}aw \'Sliwa This is my paper

Pith reviewed 2026-05-18 17:34 UTC · model grok-4.3

classification 🧮 math.FA math.GN

keywords scattered compact spacesC_p(X)complemented copies of c_0Lotz-Peck-Porta theoremRosenthal theorempointwise topologycompact spacesfunctional analysis

0 comments

The pith

An infinite compact space X is scattered exactly when every closed infinite-dimensional subspace of C_p(X) contains a complemented copy of c_0 in the pointwise topology.

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

This paper proves that for an infinite compact space X, being scattered is equivalent to every closed infinite-dimensional subspace of the pointwise function space C_p(X) containing a complemented copy of c_0. This adapts the Lotz-Peck-Porta theorem from Banach spaces to the pointwise topology setting. A reader would care because it ties the topological scatteredness property of X directly to the complemented subspace structure in its continuous functions. The work also gives a C_p analogue of Rosenthal's theorem, characterizing when C_p(X) contains uncountable copies of c_0(Gamma) by the existence of uncountable disjoint open sets in X.

Core claim

The author establishes that an infinite compact space X is scattered if and only if every closed infinite-dimensional subspace in C_p(X) contains a copy of c_0 (with the pointwise topology) which is complemented in the whole space C_p(X). This provides a C_p-version of the theorem of Lotz, Peck and Porta for Banach spaces C(X) and c_0. Additionally, for an infinite compact X the space C_p(X) contains a closed copy of c_0(Gamma) (with the pointwise topology) for some uncountable set Gamma if and only if X admits an uncountable family of pairwise disjoint open subsets of X.

What carries the argument

The complemented embedding of c_0 with the pointwise topology into closed infinite-dimensional subspaces of C_p(X), serving as a complete invariant for the scatteredness of the compact space X.

If this is right

Scattered compact spaces yield C_p(X) spaces where all infinite-dimensional closed subspaces contain complemented c_0 copies.
Non-scattered spaces must possess at least one closed infinite-dimensional subspace in C_p(X) lacking any complemented c_0 copy.
X having an uncountable family of pairwise disjoint open sets implies C_p(X) contains a closed copy of c_0(Gamma) for uncountable Gamma.
The equivalence opens applications to other C_p-theory results on subspace structures and complementation.

Where Pith is reading between the lines

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

This characterization could help construct concrete examples of function spaces exhibiting complementation failures tied to non-scatteredness.
It suggests exploring whether analogous equivalences hold when replacing the pointwise topology with the compact-open topology.
One might test the result on specific non-scattered spaces like the Stone-Cech remainder to confirm the existence of subspaces without complemented c_0.
The link to disjoint open sets may extend to classifying other topological properties detectable via c_0 embeddings in function spaces.

Load-bearing premise

The pointwise topology on C(X) makes complemented embeddings of c_0 a sufficient condition to determine that the underlying compact space X is scattered.

What would settle it

Construct or exhibit a non-scattered infinite compact space X such that every closed infinite-dimensional subspace of C_p(X) nevertheless contains a complemented copy of c_0 in the pointwise topology.

read the original abstract

For a Tychonoff space $X$ by $C_p(X)$ we denote the space $C(X)$ of continuous real valued functions on $X$ endowed with the pointwise topology. We prove that an infinite compact space $X$ is scattered if and only if every closed infinite-dimensional subspace in $C_p(X)$ contains a copy of $c_0$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$. This provides a $C_p$-version of the theorem of Lotz, Peck and Porta for Banach spaces $C(X)$ and $c_0$. Applications will be provided. We prove also a $C_p$-version of Rosenthal's theorem by showing that for an infinite compact $X$ the space $C_p(X)$ contains a closed copy of $c_{0}(\Gamma)$ (with the pointwise topology) for some uncountable set $\Gamma$ if and only if $X$ admits an uncountable family of pairwise disjoint open subsets of $X$. Illustrating examples, additional supplementing $C_p$-theorems and comments are included.

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.

Desk Editor's Note private letter to a colleague

This paper gives solid Cp-versions of the Lotz-Peck-Porta and Rosenthal theorems, with the scatteredness equivalence as the main new result.

read the letter

The main takeaway is that an infinite compact X is scattered precisely when every closed infinite-dimensional subspace of Cp(X) contains a complemented copy of c0 in the pointwise topology, and that Cp(X) contains a closed copy of c0(Gamma) for uncountable Gamma exactly when X has an uncountable family of pairwise disjoint open sets. These are presented as direct extensions of the classical Banach-space statements to the weaker pointwise setting. The extension feels natural because the pointwise topology aligns better with the topological properties of X that scatteredness and dispersion capture. The paper also supplies examples, extra Cp-theorems, and comments that help make the equivalences concrete. That supporting material is useful and shows the authors are thinking about how the results sit in the broader literature. The citation pattern is standard and appropriate, pointing back to the original Lotz-Peck-Porta and Rosenthal papers plus relevant Cp-theory work. On the soft spots, the sufficiency direction for the scattered case deserves a close look. The stress-test note correctly flags that building a continuous linear projection onto the c0 copy in the pointwise topology can be delicate when the Cantor-Bendixson derivative is uncountable; if the chosen functions lack uniformly controlled supports, the coordinate functionals may fail to be continuous on the whole space. The abstract leaves this construction implicit, so the full proof needs to demonstrate explicitly how scatteredness supplies the necessary control without extra restrictions on the subspace. If that step is handled cleanly, the central claims hold; otherwise it would be a genuine gap. Overall the arguments appear non-circular and grounded in standard techniques. This work is aimed at people who already work in Cp-theory or the classification of compact spaces via function-space properties. A reader interested in complemented subspaces or topological invariants of scattered compacta will get direct value from the equivalences. It deserves a serious referee because the statements are precise, the extensions are non-trivial, and the potential continuity issue is checkable rather than fatal.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that an infinite compact space X is scattered if and only if every closed infinite-dimensional subspace of C_p(X) contains a complemented copy of c_0 (pointwise topology) inside it. It also establishes a C_p-version of Rosenthal's theorem: C_p(X) contains a closed copy of c_0(Γ) for some uncountable Γ if and only if X has an uncountable family of pairwise disjoint nonempty open sets. The paper includes applications, examples, and supplementary C_p-results.

Significance. If the proofs hold, the result supplies a topological characterization of scatteredness via complemented c_0 subspaces in the pointwise topology, extending the Lotz-Peck-Porta theorem from the norm topology on C(X) to the C_p setting. The Rosenthal analogue similarly ties the existence of large pointwise c_0 copies to a concrete topological property of X. These equivalences could serve as useful invariants for distinguishing scattered and non-scattered compacta through function-space properties.

major comments (1)

[Main theorem on scatteredness (sufficiency proof)] The sufficiency direction (scattered X implies every closed infinite-dimensional Y ⊂ C_p(X) contains a complemented pointwise c_0) requires a continuous linear projection onto the c_0 copy. The construction via disjoint-support functions extracted from Cantor-Bendixson derivatives must be checked for pointwise continuity of the coordinate functionals when the derivative is uncountable; without uniform support control the projection may fail to be continuous on the whole C_p(X).

minor comments (2)

Clarify whether the applications are direct corollaries of the two main theorems or require additional arguments.
Ensure all statements of Rosenthal-type results explicitly record that the copy of c_0(Γ) is closed in the pointwise topology.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting a potential subtlety in the sufficiency direction of the main theorem. We address this point directly below and will incorporate clarifications into the revised version.

read point-by-point responses

Referee: [Main theorem on scatteredness (sufficiency proof)] The sufficiency direction (scattered X implies every closed infinite-dimensional Y ⊂ C_p(X) contains a complemented pointwise c_0) requires a continuous linear projection onto the c_0 copy. The construction via disjoint-support functions extracted from Cantor-Bendixson derivatives must be checked for pointwise continuity of the coordinate functionals when the derivative is uncountable; without uniform support control the projection may fail to be continuous on the whole C_p(X).

Authors: We appreciate the referee drawing attention to the need for explicit verification of continuity. In the proof, the functions are selected inductively from the Cantor-Bendixson derivatives so that their supports are pairwise disjoint. The coordinate functionals on the resulting c_0 copy are realized as point evaluations at points lying in these supports (specifically, at points where each selected function equals 1 and vanishes on all other supports). Since the pointwise topology is generated by the evaluation maps at points of X, each such functional is continuous on C_p(X). Disjointness ensures that the value at the chosen point for one coordinate is independent of the others, so no additional uniform bound on supports is required; the argument applies verbatim whether the relevant derivative level is countable or uncountable. We will revise the manuscript by adding a short paragraph immediately after the construction that spells out this continuity argument in detail. revision: yes

Circularity Check

0 steps flagged

No circularity: direct topological and functional-analytic equivalences

full rationale

The paper establishes the stated equivalences for scattered compact spaces X via standard arguments on Cantor-Bendixson derivatives, disjoint supports, and pointwise continuity of projections in C_p(X). These rest on classical results in topology and Banach space theory rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The central claims are externally falsifiable via the definitions of scatteredness and complemented subspaces, with no reduction of the target statement to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, ad-hoc axioms, or invented entities are identifiable; the work relies on standard background in topology and Cp-theory.

pith-pipeline@v0.9.0 · 5749 in / 1176 out tokens · 34220 ms · 2026-05-18T17:34:26.203641+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3: X scattered iff every closed infinite-dimensional subspace of C_p(X) contains a complemented copy of (c_0)_p
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Construction of biorthogonal system g_j, g*_j via supports on scattered derivatives

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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A. V. Arkhangel’ski,Topological Function Spaces, Kluwer, Dordrecht, 1992

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Banakh, J

T. Banakh, J. K¸ akol, W.´Sliwa,Josefson–Nissenzweig property forC p-spaces, Rev. Real Acad. Cienc. Exactas Fis. Nat.113(2019), 3015–3030

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K¸ akol, A

J. K¸ akol, A. Leiderman,A characterization ofXfor which spacesC p (X)are distinguished and its applications, Proc. Amer. Math. Soc. Series B,8(2021), 86–99

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K¸ akol, S

J. K¸ akol, S. L´ opez-Alfonso, W.´Sliwa,A characterization of scattered compact (andω-bounded) spaces, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2025)119:94

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K¸ akol, D

J. K¸ akol, D. Sobota, L. Zdomskyy,GrothendieckC(K)-spaces and the Josefson–Nissenzweig theorem, Fund. Math. 263(2), 105–131 (2023)

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K¸ akol, W.´Sliwa,Feral dual spaces and (strongly) distinguished spaces C(X)

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A. Leiderman, M. Levin, V. Pestov,On linear continuous open surjections of the spacesC p(X), Topology Appl.81(1997), 269–279

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Lindenstrauss, L

J. Lindenstrauss, L. Tzafriri,Classical Banach Spaces I, Springer 1971

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H. P. Lotz, N. T. Peck, H. Porta,Semi-embeddings of Banach spaces, Proc. Edinburgh Math. Soc.22, (1979) 233–240

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Narici, E

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Pe lczy´ nski, Z

A. Pe lczy´ nski, Z. Semadeni,Spaces of Continuous Functions III, Studia Math. 18(1958), 211–222

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[24]

G. M. Reed,On normality and countable paracompactness, Fund. Math.110 (1980), 145–152

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H. P. Rosenthal,On injective Banach spaces and the spacesL ∞(µ)for finite measuresµ, Acta Math.124(1970), 205–248

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Schwartz, ´Etude des sommes d’exponentielles r´ eelles, Paris 1943

L. Schwartz, ´Etude des sommes d’exponentielles r´ eelles, Paris 1943

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E. K. van Douwen,Orderability of all noncompact images, Topology and its Appl.51(1993), 159–172

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L. E. Ward,A generalization of the Hahn-Mazurkiewicz theorem, Proc. Am. Math. Soc.58(1976), 369–374. Faculty of Mathematics and Informatics. A. Mickiewicz University, 61-614 Pozna´n Email address:kakol@amu.edu.pl Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic Email address:kurka.ondrej@seznam.cz Faculty of Exact and Technical ...

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[1] [1]

A. V. Arkhangel’ski,Topological Function Spaces, Kluwer, Dordrecht, 1992

work page 1992

[2] [2]

Banakh, J

T. Banakh, J. K¸ akol, W.´Sliwa,Josefson–Nissenzweig property forC p-spaces, Rev. Real Acad. Cienc. Exactas Fis. Nat.113(2019), 3015–3030

work page 2019

[3] [3]

A. Dow, H. Junnila, J. Pelant,Chain conditions and weak topologies, Topology Appl.156(2009), 1327–1344

work page 2009

[4] [4]

Floret,Weakly compact sets

K. Floret,Weakly compact sets. Lecture Notes in Math.801, Springer, Berlin, 1980

work page 1980

[5] [5]

E. M. Galego, J. N. Hagler,Copies ofc 0(Γ)inC(K, X)spaces, Proc. Amer. Math. Soc.,140(2012), 3843–3852

work page 2012

[6] [6]

Haydon,On a problem of Pe lczy´ nski; Milutin spaces, Dugundji spaces and AE(0−dim), Studia Math

R. Haydon,On a problem of Pe lczy´ nski; Milutin spaces, Dugundji spaces and AE(0−dim), Studia Math. T.LII (1974), 23–31

work page 1974

[7] [7]

Jarchow, H.,Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981

work page 1981

[8] [8]

K¸ akol, W

J. K¸ akol, W. Kubi´ s, M. L´ opez-Pellicer, D. Sobota,Descriptive Topology in Se- lected Topics of Functional Analysis.Updated and Expanded Second Edition, Springer, Developements in Math.24, New York Dordrecht Heidelberg, 2025

work page 2025

[9] [9]

K¸ akol, O

J. K¸ akol, O. Kurka,A new characterization of compact scattered spacesXin terms of spacesC p(X), Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2025) 119:43

work page 2025

[10] [10]

K¸ akol, A

J. K¸ akol, A. Molt´ o, W.´Sliwa,On subspaces of spacesC p (X)isomorphic to spacesc 0 andℓ q with the topology induced fromR N, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023)117:154

work page 2023

[11] [11]

K¸ akol, A

J. K¸ akol, A. Leiderman,On linear continuous operators between distinguished spacesC p(X), Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2021), 115:199

work page 2021

[12] [12]

K¸ akol, A

J. K¸ akol, A. Leiderman,When is a Locally Convex Space Eberlein- Grothendieck?, Results Math (2022) 77:236

work page 2022

[13] [13]

K¸ akol, A

J. K¸ akol, A. Leiderman,A characterization ofXfor which spacesC p (X)are distinguished and its applications, Proc. Amer. Math. Soc. Series B,8(2021), 86–99

work page 2021

[14] [14]

K¸ akol, S

J. K¸ akol, S. L´ opez-Alfonso, W.´Sliwa,A characterization of scattered compact (andω-bounded) spaces, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2025)119:94

work page 2025

[15] [15]

K¸ akol, D

J. K¸ akol, D. Sobota, L. Zdomskyy,GrothendieckC(K)-spaces and the Josefson–Nissenzweig theorem, Fund. Math. 263(2), 105–131 (2023)

work page 2023

[16] [16]

K¸ akol, W.´Sliwa,Feral dual spaces and (strongly) distinguished spaces C(X)

J. K¸ akol, W.´Sliwa,Feral dual spaces and (strongly) distinguished spaces C(X). Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A-Mat. RACSAM 117 (2023), no. 3, Paper No. 94, 15 pp

work page 2023

[17] [17]

R. W. Knight, ∆-Sets, Trans. Amer. Math. Soc.339(1993), 45–60

work page 1993

[18] [18]

Leiderman, M

A. Leiderman, M. Levin, V. Pestov,On linear continuous open surjections of the spacesC p(X), Topology Appl.81(1997), 269–279

work page 1997

[19] [19]

Lindenstrauss, L

J. Lindenstrauss, L. Tzafriri,Classical Banach Spaces I, Springer 1971

work page 1971

[20] [20]

H. P. Lotz, N. T. Peck, H. Porta,Semi-embeddings of Banach spaces, Proc. Edinburgh Math. Soc.22, (1979) 233–240

work page 1979

[21] [21]

H. P. Lotz, N. T. Peck, H. Porta,Semiembeddings in Topological Linear Spaces, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983. 749–763

work page 1981

[22] [22]

Narici, E

L. Narici, E. Beckeinstain,Topological Vector Spaces, Pure and Applied Math- ematics, Marcel Dekker, INC. 1985. 20 JERZY KA ¸ KOL, OND ˇREJ KURKA, AND WIES LAW ´SLIWA

work page 1985

[23] [23]

Pe lczy´ nski, Z

A. Pe lczy´ nski, Z. Semadeni,Spaces of Continuous Functions III, Studia Math. 18(1958), 211–222

work page 1958

[24] [24]

G. M. Reed,On normality and countable paracompactness, Fund. Math.110 (1980), 145–152

work page 1980

[25] [25]

H. P. Rosenthal,On injective Banach spaces and the spacesL ∞(µ)for finite measuresµ, Acta Math.124(1970), 205–248

work page 1970

[26] [26]

Schwartz, ´Etude des sommes d’exponentielles r´ eelles, Paris 1943

L. Schwartz, ´Etude des sommes d’exponentielles r´ eelles, Paris 1943

work page 1943

[27] [27]

E. K. van Douwen,Orderability of all noncompact images, Topology and its Appl.51(1993), 159–172

work page 1993

[28] [28]

L. E. Ward,A generalization of the Hahn-Mazurkiewicz theorem, Proc. Am. Math. Soc.58(1976), 369–374. Faculty of Mathematics and Informatics. A. Mickiewicz University, 61-614 Pozna´n Email address:kakol@amu.edu.pl Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic Email address:kurka.ondrej@seznam.cz Faculty of Exact and Technical ...

work page 1976