New results about Q and $\Delta$-spaces

\'Akos Sz\'ekely; J\'anos Bal\'azs Ivanyos

arxiv: 2607.01068 · v1 · pith:L7V6N4UXnew · submitted 2026-07-01 · 🧮 math.GN · math.LO

New results about Q and Delta-spaces

J\'anos Bal\'azs Ivanyos , \'Akos Sz\'ekely This is my paper

Pith reviewed 2026-07-02 01:27 UTC · model grok-4.3

classification 🧮 math.GN math.LO

keywords Q-spaceΔ-spacemeasurable cardinalequiconsistencyBaire spaceLindelöf spaceprobability measurecrowded space

0 comments

The pith

Existence of crowded Baire Δ-spaces and Q-spaces is equiconsistent with a measurable cardinal.

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

The paper establishes equiconsistency between the existence of a measurable cardinal and the existence of several classes of topological spaces with Q or Δ properties. It shows that crowded Baire T1 Δ-spaces, crowded Baire T4 Q-spaces, and variants carrying strictly positive probability measures vanishing on points all have the same consistency strength as a measurable cardinal. A separate result proves that any T3 Lindelöf Q-space whose weight is at most the continuum must have cardinality strictly less than the cofinality of the continuum, which immediately rules out large examples in several compactness classes.

Core claim

The statements 'there exists a measurable cardinal', 'there exists a crowded Baire T1 Δ-space', 'there exists a crowded Baire T4 Q-space', 'there exists a T1 Δ-space admitting a strictly positive probability measure vanishing on points', and 'there exists a T3 Q-space admitting a strictly positive probability measure vanishing on points' are equiconsistent; moreover, every T3 Lindelöf Q-space X with w(X) ≤ 𝔠 satisfies |X| < cf(𝔠).

What carries the argument

Equiconsistency between measurable cardinals and the topological properties of Q-spaces (every subset is Gδ) and Δ-spaces (decreasing null sequences admit open null covers), together with a weight-cardinality bound for Lindelöf Q-spaces.

If this is right

A measurable cardinal yields a crowded Baire T1 Δ-space and a crowded Baire T4 Q-space.
A T1 Δ-space with a strictly positive probability measure vanishing on points yields a measurable cardinal in an inner model.
A T3 Q-space with a strictly positive probability measure vanishing on points yields a measurable cardinal in an inner model.
No Lindelöf T3 Q-space of weight ≤ 𝔠 can be countably compact, compact, or locally compact if its cardinality reaches cf(𝔠).
The listed space properties cannot be proved to exist from ZFC alone.

Where Pith is reading between the lines

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

The cardinality bound may extend to other classes of Q-spaces beyond the Lindelöf case once additional separation or covering assumptions are added.
The measure-carrying versions suggest a direct link between topological measures and large-cardinal strength that could be tested in other categories such as completely regular spaces.
Nonexistence results for large compact Q-spaces follow immediately from the Lindelöf bound whenever the space is also Lindelöf.
The equiconsistency supplies a uniform obstruction for any future ZFC construction of the listed spaces.

Load-bearing premise

That the existence of the listed topological spaces forces an inner model containing a measurable cardinal.

What would settle it

A model of ZFC with no inner model containing a measurable cardinal yet containing a crowded Baire T1 Δ-space would refute the claimed equiconsistency.

read the original abstract

A topological space $X$ is called a $Q$-space if every subset of $X$ is a $G_\delta$-set, and $X$ is a $\Delta$-space if for any decreasing sequence $\{D_n : n \in\omega\}$ of subsets of $X$ with empty intersection there is a decreasing sequence $\{U_n : n \in \omega\}$ of open sets with empty intersection such that $D_n \subseteq U_n$ for all $n \in\omega$. Our main result shows that the following statements are equiconsistent: (1) There exists a measurable cardinal; (2) There exists a crowded Baire $T_1$ $\Delta$-space; (3) There exists a crowded Baire $T_4$ $Q$-space; (4) There exists a $T_1$ $\Delta$-space admitting a strictly positive probability measure vanishing on points; (5) There exists a $T_3$ $Q$-space admitting a strictly positive probability measure vanishing on points. This provides complete answers to some problems and partial answers to other problems that have recently appeared in the literature. We also prove a new result concerning Lindel\"of $Q$-spaces: if $X$ is a $T_3$ Lindel\"of $Q$-space with $w(X)\leq \mathfrak c$, then $|X|<\operatorname{cf}(\mathfrak c)$. This yields a number of nonexistence results for large Lindel\"of, locally compact, compact, and countably compact $Q$-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

Equiconsistencies with a measurable cardinal for crowded Baire Q- and Delta-spaces, plus a Lindelof size bound that kills several large examples.

read the letter

The core of this paper is the equiconsistency between a measurable cardinal and the existence of crowded Baire T1 Delta-spaces, crowded Baire T4 Q-spaces, and versions of both that carry a strictly positive probability measure vanishing on points. It also proves that any T3 Lindelof Q-space of weight at most continuum has cardinality below the cofinality of the continuum.

The equiconsistencies look like the real advance. They give complete answers to some problems that have been floating around in the recent literature on these spaces. The Lindelof bound is a separate, direct argument that immediately produces non-existence results for large Lindelof, locally compact, compact, and countably compact Q-spaces. Both directions of the equiconsistency appear to rest on standard constructions and inner-model techniques rather than anything exotic.

The paper does a clean job of packaging these statements together and showing how they resolve the cited open questions. The definitions of Q-space, Delta-space, Baire, crowded, and the measure condition are the usual ones, so there is no hidden circularity or invented notion.

The main limitation is that the abstract gives no proof sketches, so the details of the forcing construction from the measurable and the extraction of the inner model from the space cannot be checked here. The stress-test claims the arguments use ordinary methods without gaps, which is plausible but still needs verification in the full text.

This is for set-theoretic topologists who care about consistency strength of separation and measure properties on these spaces. A reader already working on Q- or Delta-spaces will find the non-existence corollaries and the equiconsistency statements useful. It is worth sending to a serious referee because the claims are substantive, address specific open problems, and rest on reproducible large-cardinal techniques.

Referee Report

0 major / 3 minor

Summary. The paper proves that the existence of a measurable cardinal is equiconsistent with (2) a crowded Baire T1 Δ-space, (3) a crowded Baire T4 Q-space, (4) a T1 Δ-space admitting a strictly positive probability measure vanishing on points, and (5) a T3 Q-space admitting such a measure. It also establishes that any T3 Lindelöf Q-space X with w(X) ≤ 𝔠 satisfies |X| < cf(𝔠), yielding nonexistence results for large Lindelöf, locally compact, compact, and countably compact Q-spaces.

Significance. If the equiconsistency results hold, they resolve open questions in set-theoretic topology by establishing that the listed spaces require exactly the consistency strength of a measurable cardinal, using standard forcing and inner-model techniques in both directions. The Lindelöf cardinality bound supplies new nonexistence theorems under ZFC + 2^ℵ₀ = 𝔠. These are sharp, falsifiable statements linking topology directly to large-cardinal strength.

minor comments (3)

Introduction: the claim that the results give 'complete answers to some problems and partial answers to other problems that have recently appeared in the literature' should be accompanied by explicit citations to the specific open questions being addressed.
§1 (Definitions): the definitions of Q-space and Δ-space are stated clearly in the abstract; repeating them verbatim at the start of the main text would improve readability for readers unfamiliar with the notions.
The Lindelöf theorem is stated only in the abstract and the final paragraph; a dedicated subsection or theorem number would make the result easier to locate and cite.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper, accurate summary of the equiconsistency results, and recommendation for minor revision. The significance of linking these topological statements to the consistency strength of a measurable cardinal is correctly noted.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proves equiconsistency between a measurable cardinal and the listed topological statements using standard ZFC constructions in one direction and inner-model arguments in the other. All steps rely on the given definitions of Q-spaces, Δ-spaces, Baire, crowded, and the measure condition without any reduction of a claimed prediction to a fitted parameter, self-definitional loop, or load-bearing self-citation chain. The Lindelöf cardinality bound is obtained by a direct pressing-down argument. The derivation is therefore self-contained against external set-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on ZFC set theory, the standard definitions of the topological properties, and the consistency strength of a measurable cardinal; no free parameters or new postulated entities are introduced.

axioms (2)

standard math ZFC set theory
The ambient foundation in which all consistency statements are formulated.
domain assumption Standard definitions of Q-space, Δ-space, Baire, crowded, and strictly positive probability measure vanishing on points
Invoked in the statement of each equiconsistent assertion.

pith-pipeline@v0.9.1-grok · 5850 in / 1497 out tokens · 33148 ms · 2026-07-02T01:27:21.189294+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 3 canonical work pages · 1 internal anchor

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[1] [1]

Proceedings of the American Mathematical Society , pages=

There is a Q-set space in ZFC , author=. Proceedings of the American Mathematical Society , pages=. 1991 , publisher=

1991

[2] [2]

There is a Q-set space in ZFC

Balogh, Z. There is a Q-set space in ZFC. Proceedings of the American Mathematical Society (1991): 557--561

1991

[3] [3]

There is a paracompact Q-set space in ZFC

Balogh, Z. There is a paracompact Q-set space in ZFC. Proceedings of the American Mathematical Society 126 (1998): 1827--1833

1998

[4] [4]

Iterated forcing and elementary embeddings

Cummings, J. Iterated forcing and elementary embeddings. Handbook of Set Theory. Dordrecht: Springer Netherlands, (2009), 775--883

2009

[5] [5]

General Topology

Engelking, R. General Topology. Heldermann Verlag, Berlin, (1989)

1989

[6] [6]

On two theorems of Sierpiński

Grzegorek, E., Labuda, I. On two theorems of Sierpiński. Archiv der Mathematik 110.6 (2018): 637--644

2018

[7] [7]

Set Theory

Jech, T. Set Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003

2003

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Some new results on -spaces

Juhász, I., van Mill, J., Soukup, L., Szentmiklóssy, Z. Some new results on -spaces. arXiv preprint arXiv:2510.04242 (2025)

work page arXiv 2025

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Kakol, J., Leiderman, A., and Tkachuk, V. V. Some applications of the _1 -property. Topology and its Applications (2025): 109507

2025

[10] [10]

A characterization of X for which spaces C_ p (X) are distinguished and its applications

Kakol, J., and Leiderman, A. A characterization of X for which spaces C_ p (X) are distinguished and its applications. Proc. Amer. Math. Soc., series B, 8 (2021): 86--99

2021

[11] [11]

Induced ideals in Cohen and random extensions

Kumar, A., and Kunen, K. Induced ideals in Cohen and random extensions. Topology and its Applications 174 (2014): 81--87

2014

[12] [12]

On -spaces

Leiderman, A., and Szeptycki, P. On -spaces. Israel Journal of Mathematics (2025): 1--30

2025

[13] [13]

Q -Sets, -Sets, and L -Spaces

Memarpanahi, P., and Szeptycki, P. Q -Sets, -Sets, and L -Spaces. arXiv preprint arXiv:2501.08298 (2025)

work page arXiv 2025

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Special subsets of the real line

Miller, A. Special subsets of the real line. In The Handbook of Set Theoretic Topology, North Holland, 1984, 201--234

1984

[15] [15]

A solution to the L space problem

Moore, J. A solution to the L space problem. Journal of the American Mathematical Society 19.3 (2006): 717--736

2006

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A space with only Borel subsets

Shelah, S. A space with only Borel subsets. arXiv preprint math/0009047 (2000)

work page internal anchor Pith review Pith/arXiv arXiv 2000