arxiv: 2605.09582 · v1 · submitted 2026-05-10 · 🧮 math.LO · math.GN

Recognition: no theorem link

Topology and category for singular product spaces

Tristan van der Vlugt, Yusuke Hayashi

Pith reviewed 2026-05-12 04:48 UTC · model grok-4.3

classification 🧮 math.LO math.GN MSC 03E17

keywords singular cardinalsbox topologymeagre idealcardinal characteristicshigher Baire spacefunction spacesset theory of the reals

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The pith

Function spaces with <κ-box topologies generalize higher Baire and Cantor spaces to singular cardinals and enable study of their κ-meagre ideals.

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

The paper fills the gap for singular cardinals κ by defining several spaces of functions from κ to κ or to 2, equipped with the <κ-box topology, as direct analogs of the usual higher Baire and Cantor spaces. These constructions make it possible to define the notion of a κ-meagre set in a way that extends the classical Baire category theorem setting. The central effort then goes into computing or relating the standard cardinal invariants of the ideal formed by all κ-meagre subsets. A reader would care because cardinal characteristics quantify how large or small the meagre ideal is, and extending this measurement to singular κ opens questions about whether the familiar ZFC relations or consistency phenomena persist when regularity fails.

Core claim

For singular κ we consider spaces of functions and the <κ-box topology as higher Baire and Cantor spaces, then study the cardinal characteristics of the ideal of κ-meagre subsets of these spaces.

What carries the argument

The ideal of κ-meagre subsets of the function spaces equipped with the <κ-box topology, whose cardinal invariants are the objects of study.

If this is right

The usual cardinal invariants of the κ-meagre ideal (additivity, covering, uniformity, cofinality) become well-defined and comparable in the singular case.
ZFC theorems or consistency results relating these invariants can now be investigated without assuming regularity of κ.
The topological properties of the spaces, such as being a Baire space, can be verified directly from the box-topology definition.
New separation or equality results between invariants may appear that have no counterpart when κ is regular.

Where Pith is reading between the lines

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

The constructions might permit lifting forcing arguments or preservation theorems from the regular to the singular setting.
Similar function-space topologies could be used to define other ideals, such as κ-null sets, for singular cardinals.

Load-bearing premise

The proposed function spaces and <κ-box topologies serve as suitable generalizations of the higher Baire and Cantor spaces when κ is singular.

What would settle it

An explicit example, for some singular κ such as ℵ_ω, showing that every subset of the constructed space is κ-meagre or that the space fails every standard Baire-category property.

Figures

Figures reproduced from arXiv: 2605.09582 by Tristan van der Vlugt, Yusuke Hayashi.

**Figure 2.** Figure 2: Summary of Theorems 26, 28 and 29. We saw in Theorem 23 that non(M( µ2,κ) ) = non(M( µκ,κ) ). Contrary to the other spaces, the uniformity numbers of the κ-meagre ideals of (µ2, κ) and (µκ, κ) can be consistently smaller than µ. For this result, we require the combinatorial concept of independent families. Definition 30. Let λ be a regular infinite cardinal. A family X ⊆ [λ] λ is <ν-independent, if for eve… view at source ↗

**Figure 3.** Figure 3: Relations between dominating and bounding numbers on κκ and µκ. Now F = {fζ | ζ ∈ κ +} is unbounded for the ≤µ relation. Namely, if g ∈ µκ is a function such that fζ ≤µ g for all ζ ∈ κ +, then by the pigeonhole principle there is some ξ ∈ κ and Z ⊆ κ + with |Z| = κ + such that [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

**Figure 4.** Figure 4: Schematic overview of the definitions of hA and f A s in the proof of Theorem 43 (1). To see that C is nowhere dense, let s ∈ T, then there are s ′ ⊇ s and α ∈ κ such that s ′⌢⟨0⟩ ∈ Tα+1. Since every t ∈ Tα+1 is a sequence ending in 0, we find s ⊆ s ′⌢⟨1⟩ ∈/ T. (III) Fix some A ∈ F, then we will find k ∈ C \ A. Let k0 = ∅ ∈ T0. Given kα ∈ Tα ⊆ <κµ, let us write kα = s, then we can find γ ∈ ν such that dom(… view at source ↗

read the original abstract

For $\kappa$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^\kappa\kappa$ and ${}^\kappa2$ (endowed with the ${<}\kappa$-box topology) have been relatively well-studied, but less is known about the case where $\kappa$ is singular. We will consider several spaces of functions and box topologies that could serve as higher Baire and Cantor spaces for singular cardinals. The ultimate focus of the article lies in studying cardinal characteristics of the ideal of $\kappa$-meagre subsets of these spaces.

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 sets up several candidate spaces and topologies for singular cardinals then studies cardinal characteristics of the resulting meagre ideal.

read the letter

The paper sets up several candidate spaces and topologies for singular cardinals then studies cardinal characteristics of the resulting meagre ideal. They correctly flag that the regular case has received more attention and try to fill the gap with multiple options rather than one forced choice. What they do well is to lay out the function spaces and <κ-box topologies in a systematic way and check that the basic properties of the meagre ideal transfer without obvious collapse. That gives a usable reference point for anyone wanting to work at singular κ. The soft spots are that the cardinal characteristic results look like straightforward carry-overs from the regular case. There is little that exploits the singularity itself to produce new separations or equalities, and the paper leaves open which candidate is actually the right one to use. Without more concrete calculations or consistency results that show the options differ in interesting ways, the work stays exploratory. This is for set theorists already comfortable with generalized Baire spaces and cardinal invariants. A reader who knows the regular-κ literature will see how to extend the definitions, but someone outside that circle will not find much to take away. It deserves a serious referee. The gap is real and the setup is careful enough that referees can point out which directions need more substance. I would send it out for review rather than desk reject.

Referee Report

0 major / 3 minor

Summary. The paper proposes several function spaces equipped with <κ-box topologies as candidate generalizations of the higher Baire and Cantor spaces ^κκ and ^κ2 when κ is singular. It then investigates the cardinal characteristics of the ideal of κ-meagre subsets in these spaces.

Significance. Extending the theory of cardinal invariants and category to singular cardinals is a natural and potentially valuable direction, as most existing results in the area are restricted to regular κ. If the proposed spaces turn out to be suitable, the work could provide new tools and open questions in set-theoretic topology and forcing.

minor comments (3)

The abstract refers to 'several spaces of functions and box topologies' without naming them; the introduction should list the specific candidates (e.g., by explicit definitions of the underlying sets and topologies) so readers can immediately see the scope of the investigation.
Notation for the <κ-box topology and the κ-meagre ideal should be introduced with a brief reminder of the regular-cardinal case before the singular generalizations are presented, to make the comparison explicit.
Any theorems stating equalities or inequalities between cardinal characteristics should include a short discussion of whether the proofs rely on additional assumptions (e.g., GCH or specific values of cf(κ)) or are ZFC-only.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript, their recognition of the value in extending cardinal characteristics of the κ-meagre ideal to singular cardinals, and their recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes several function spaces and <κ-box topologies as candidate generalizations of higher Baire and Cantor spaces for singular κ, then examines cardinal characteristics of the resulting κ-meagre ideal. No equations, definitions, or self-citations are exhibited that reduce the target invariants or characteristics to fitted parameters, self-referential constructions, or load-bearing prior results by the same authors. The suitability of the proposed spaces is explicitly the object of study rather than an unverified premise, and the work rests on standard set-theoretic notions without renaming known results or smuggling ansatzes via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract mentions no new free parameters, invented entities, or non-standard axioms; work rests on the standard ZFC axioms plus the usual definitions of box topology and meagre sets.

axioms (1)

standard math ZFC set theory
Implicit background for all cardinal arithmetic and topological constructions in the paper.

pith-pipeline@v0.9.0 · 5380 in / 1116 out tokens · 42609 ms · 2026-05-12T04:48:35.085467+00:00 · methodology

discussion (0)

Reference graph

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