Generalized almost disjoint families and injective Banach spaces

Chris Lambie-Hanson; David Schrittesser

arxiv: 2606.02040 · v1 · pith:BLUBCCDGnew · submitted 2026-06-01 · 🧮 math.FA · math.LO

Generalized almost disjoint families and injective Banach spaces

Chris Lambie-Hanson , David Schrittesser This is my paper

Pith reviewed 2026-06-28 12:34 UTC · model grok-4.3

classification 🧮 math.FA math.LO

keywords almost disjoint familiesinjective dimensionBanach space c0continuum hypothesisCech-Stone remainderhomological theorybounding number

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The pith

If the continuum hypothesis holds, the injective dimension of c0 is at least 3.

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

The paper works to establish a concrete lower bound on the injective dimension of the Banach space c0, a quantity whose exact value remains unknown in general. To reach this bound the authors generalize the classical notion of an almost disjoint family from subsets of the naturals to families on an arbitrary topological space, and they prove that when the bounding number equals the continuum there is such a family of size 2 to the aleph-1 on the Cech-Stone remainder of the naturals. Under the continuum hypothesis this size is large enough to force any injective resolution of c0 to have length at least three. A sympathetic reader would care because the result supplies the first known lower bound of three for this dimension under a familiar set-theoretic assumption.

Core claim

Under the continuum hypothesis the injective dimension of c0 is at least 3. The proof proceeds by first introducing almost disjoint families on topological spaces and then showing that, whenever the bounding number equals the continuum, an almost disjoint family of cardinality 2 to the aleph-1 exists on the Cech-Stone remainder of the naturals; this family is used to construct a long exact sequence that witnesses the required dimension.

What carries the argument

A generalized almost disjoint family on a topological space X, which extends the classical notion on the naturals and is used to produce long injective resolutions of c0.

If this is right

Under the continuum hypothesis any injective resolution of c0 must have length at least three.
When the bounding number equals the continuum there exists an almost disjoint family of size 2 to the aleph-1 on the Cech-Stone remainder of the naturals.
The generalized notion of almost disjoint families on topological spaces is sufficient to produce new lower bounds in the homological theory of Banach spaces.

Where Pith is reading between the lines

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

The same construction might yield lower bounds for injective dimensions of other classical Banach spaces once suitable almost disjoint families are found on their associated remainders.
If the continuum hypothesis fails the dimension of c0 could drop to 2, suggesting the exact value is sensitive to the underlying set theory.
The generalized families may connect to questions about the structure of beta N minus N beyond the single application to c0.

Load-bearing premise

The proof requires both the continuum hypothesis and the existence of an almost disjoint family of size 2 to the aleph-1 on the Cech-Stone remainder when the bounding number equals the continuum.

What would settle it

A model of set theory in which the continuum hypothesis holds yet the injective dimension of c0 equals 2, or a direct computation showing that no almost disjoint family of the stated size exists on the remainder when the bounding number equals the continuum.

read the original abstract

A fundamental open problem in the homological theory of Banach spaces is the calculation of the injective dimension of the Banach space $c_0$. We make a contribution to the study of this problem by proving that, if the Continuum Hypothesis ($\mathsf{CH}$) holds, then the injective dimension of $c_0$ is at least 3. In the course of proving this result, we introduce the notion of an \emph{almost disjoint family} on a topological space $X$, generalizing the classical notion of almost disjoint families of subsets of $\mathbb{N}$, which we feel is of interest in its own right. We prove that, if $\mathfrak{b} = 2^{\aleph_0}$, then there exists an almost disjoint family of cardinality $2^{\aleph_1}$ on the \v{C}ech-Stone remainder of $\mathbb{N}$.

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.

Desk Editor's Note private letter to a colleague

The paper shows under CH that the injective dimension of c0 is at least 3 via a new generalization of almost disjoint families to topological spaces.

read the letter

The paper's key contribution is proving that the Continuum Hypothesis implies the injective dimension of c0 is at least 3. Along the way they define almost disjoint families for arbitrary topological spaces and construct one of size 2^aleph1 on beta N minus N assuming b equals the continuum. This seems like a genuine extension rather than a minor tweak to existing notions.

What the paper does well is tie the set-theoretic construction directly to the Banach space question. The lower bound is meaningful within the subfield even though it's conditional. The stress test didn't find inconsistencies in the hypotheses, which is good.

Soft spots are minor: the result is explicitly conditional, so its applicability depends on whether one accepts CH or works in models where it holds. The existence of the family relies on b = 2^aleph0, which CH forces but could be explored further in other models. Since the full proof isn't in the abstract, one would want to verify the details of how the family induces the dimension lower bound, but nothing suggests a circularity or invented entities beyond the stated generalization.

This is for researchers in Banach space homology or set-theoretic methods in analysis. A reader who follows papers on injective dimensions or almost disjoint families would get something concrete from it. It deserves serious refereeing because it addresses a fundamental open invariant with a new tool.

Recommendation: send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that if the Continuum Hypothesis holds, then the injective dimension of the Banach space c_0 is at least 3. The argument proceeds by introducing the notion of an almost disjoint family on a general topological space X (generalizing the classical notion on N), and by establishing that the assumption b = 2^aleph_0 implies the existence of such a family of cardinality 2^aleph_1 on beta N minus N; under CH the cardinal arithmetic hypothesis holds and yields the stated lower bound on injective dimension.

Significance. The result supplies a conditional lower bound for a long-standing open question in the homological theory of Banach spaces. The auxiliary construction of large almost disjoint families on the Čech-Stone remainder is of potential independent interest in set-theoretic topology and may find further applications. The paper explicitly credits the set-theoretic hypotheses and presents the main theorem as conditional rather than absolute.

minor comments (3)

[Abstract] The abstract states the main theorem clearly but does not indicate the precise location in the text where the reduction from the almost disjoint family to the injective-dimension lower bound is carried out; a forward reference would help readers.
[Definition section] Notation for the generalized almost disjoint family (Definition in the early sections) is introduced without an immediate comparison table to the classical case on N; adding such a table would improve readability.
[Existence theorem] The proof that b = 2^aleph_0 yields an almost disjoint family of size 2^aleph_1 on beta N \ N is presented in a single block; breaking it into numbered steps or lemmas would make the dependence on the bounding number more transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The report contains no specific major comments to address point by point.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation establishes a conditional lower bound on the injective dimension of c0 under CH by first defining generalized almost disjoint families on topological spaces (a new notion) and then proving, from the assumption b = 2^aleph0, the existence of such a family of size 2^aleph1 on beta N ∖ N; this family is then used to obtain the dimension bound. No step reduces by definition or by self-citation to the target conclusion, and the central argument is self-contained against the stated set-theoretic hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on two set-theoretic assumptions and the introduction of one new concept; no numerical free parameters appear.

axioms (2)

domain assumption Continuum Hypothesis (CH)
Invoked to obtain the lower bound of 3 on the injective dimension of c0.
domain assumption b = 2^aleph0
Used to guarantee an almost disjoint family of size 2^aleph1 on the Cech-Stone remainder.

invented entities (1)

almost disjoint family on a topological space X no independent evidence
purpose: Generalizes the classical notion on N to prove the main result about injective dimension.
New definition introduced in the paper; no independent evidence outside the construction is provided.

pith-pipeline@v0.9.1-grok · 5671 in / 1318 out tokens · 34134 ms · 2026-06-28T12:34:58.671608+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 2 linked inside Pith

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2013

[3] [3]

Antonio Avil´ es, F´ elix Cabello S´ anchez, Jes´ us M. F. Castillo, Manuel Gonz´ alez, and Yolanda Moreno.Separably injective Banach spaces, volume 2132 ofLecture Notes in Mathematics. Springer, [Cham], 2016

2016

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Cabello S´ anchez, J

F. Cabello S´ anchez, J. M. F. Castillo, and R. Garc´ ıa. Homological dimensions of Banach spaces.Mat. Sb., 212(4):91–112, 2021

2021

[5] [5]

Condensed mathematics and complex geometry, 2026

Dustin Clausen and Peter Scholze. Condensed mathematics and complex geometry, 2026. arXiv preprint: 2605.11731

Pith/arXiv arXiv 2026

[6] [6]

Almost disjoint families and topology

Michael Hruˇ s´ ak. Almost disjoint families and topology. InRecent progress in general topology. III, pages 601–638. Atlantis Press, Paris, 2014

2014

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R. S. Phillips. On linear transformations.Trans. Amer. Math. Soc., 48:516–541, 1940

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Rosenthal

Haskell P. Rosenthal. On relatively disjoint families of measures, with some applications to Banach space theory.Studia Math., 37:13–36, 1970

1970

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Personal communication, 2019

Peter Scholze. Personal communication, 2019

2019

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Lectures on condensed mathematics, 2026

Peter Scholze. Lectures on condensed mathematics, 2026. arXiv preprint: 2605.03658

Pith/arXiv arXiv 2026

[11] [11]

Projection of the space (m) on its subspace (c 0).Bull

Andrew Sobczyk. Projection of the space (m) on its subspace (c 0).Bull. Amer. Math. Soc., 47:938–947, 1941

1941

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Mathematical Notes: Projectingmontoc 0.Amer

Robert Whitley. Mathematical Notes: Projectingmontoc 0.Amer. Math. Monthly, 73(3):285–286, 1966. GENERALIZED ALMOST DISJOINT F AMILIES AND INJECTIVE BANACH SPACES 11 (Lambie-Hanson)Institute of Mathematics, Czech Academy of Sciences, ˇZitn´a 25, 110 00 Praha 1, Czech Republic Email address:lambiehanson@math.cas.cz URL:https://clambiehanson.github.io/ (Sch...

1966