On some relations between properties of invariant $\sigma$-ideals in Polish spaces

Marcin Michalski

arxiv: 1907.09306 · v1 · pith:YN5JHVZInew · submitted 2019-07-18 · 🧮 math.GN

On some relations between properties of invariant σ-ideals in Polish spaces

Marcin Michalski This is my paper

Pith reviewed 2026-05-24 19:27 UTC · model grok-4.3

classification 🧮 math.GN

keywords invariant sigma-idealsPolish spacesc-cc propertyWeaker Smital PropertyFubini Propertytall idealsLuzin setsnonmeasurability

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The pith

c-cc σ-ideals are tall and the Weaker Smital Property forces every Borel I-positive set to contain a non(I) witness while satisfying ccc and Fubini.

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

This paper examines relations among properties of invariant σ-ideals on Polish spaces. It establishes that any σ-ideal satisfying the c-cc condition must be tall. It shows that the presence of the Weaker Smital Property ensures every Borel set that is positive for the ideal contains a witness for the cardinal non(I) and itself satisfies both the countable chain condition and the Fubini property. The paper supplies a characterization of when I-Luzin sets are nonmeasurable and proves that the ideal of at most countable subsets of the reals fails to have the Fubini property.

Core claim

The paper proves that every c-cc σ-ideal is tall. It shows that if an invariant σ-ideal has the Weaker Smital Property, then every Borel I-positive set contains a witness for non(I), satisfies the ccc, and has the Fubini Property. It gives a characterization of the nonmeasurability of I-Luzin sets. It proves that the ideal [R]≤ω does not possess the Fubini Property, using a lemma about perfect sets.

What carries the argument

Invariant σ-ideals on Polish spaces together with the implications linking c-cc, tallness, the Weaker Smital Property, ccc, and the Fubini Property.

Load-bearing premise

The σ-ideals under study are invariant under translations or continuous maps.

What would settle it

A c-cc invariant σ-ideal that is not tall, or a Weaker Smital ideal together with a Borel I-positive set that lacks a non(I) witness, would refute the main claims.

read the original abstract

In this paper we shall consider a couple of properties of $\sigma$-ideals and study relations between them. Namely we will prove that $\mathfrak{c}$-cc $\sigma$-ideals are tall and that the Weaker Smital Property implies that every Borel $\mathcal{I}$-positive set contains a witness for non($\mathcal{I}$) as well, as satisfying ccc and Fubini Property. We give also a characterization of nonmeasurability of $\mathcal{I}$-Luzin sets and prove that the ideal $[\mathbb{R}]^{\leq\omega}$ does not posses the Fubini Property using some interesting lemma about perfect sets.

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

Paper shows c-cc ideals are tall and ties Weaker Smital Property to ccc, Fubini and non(I) witnesses, plus a characterization and counterexample, but all within a narrow subfield.

read the letter

The main points are that c-cc sigma-ideals are tall, the Weaker Smital Property forces every Borel I-positive set to contain a non(I) witness while also satisfying ccc and Fubini, there's a characterization of nonmeasurability for I-Luzin sets, and [R]<=omega fails Fubini via a lemma on perfect sets. These are presented as direct implications for invariant sigma-ideals on Polish spaces. The paper does a clean job stating and proving these specific links, and the perfect-set lemma for the counterexample is a useful concrete tool. It organizes relations that specialists might want recorded. The results are incremental rather than transformative. They fill gaps in the existing web of ideal properties without resolving open questions or introducing methods that travel outside this corner of descriptive set theory. The abstract gives no indication of heavy overlap with prior work, but the claims themselves look like straightforward extensions of known definitions. This is for readers already deep in sigma-ideals, Smital properties, and combinatorial properties of Polish spaces. A broader audience in set theory or topology will find little to use. The work is formally grounded enough and the statements precise enough that it deserves a serious referee to check the derivations and novelty against the literature. I would send it out for review rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The paper examines relations among properties of invariant σ-ideals on Polish spaces. It proves that every c-cc σ-ideal is tall, that the Weaker Smital Property implies every Borel I-positive set contains a non(I)-witness and satisfies both the ccc and the Fubini Property, supplies a characterization of nonmeasurability for I-Luzin sets, and shows via a lemma on perfect sets that the ideal [ℝ]≤ω fails the Fubini Property.

Significance. If the proofs are correct, the results link several standard properties (tallness, c-cc, Weaker Smital, ccc, Fubini, nonmeasurability) for invariant σ-ideals, which are frequently used in forcing and descriptive set theory. The explicit counter-example for [ℝ]≤ω and the characterization of I-Luzin nonmeasurability are concrete contributions that may be cited in subsequent work on cardinal invariants of the continuum.

minor comments (2)

Abstract: the clause 'as well, as satisfying ccc and Fubini Property' is grammatically awkward and should be rephrased for readability.
Abstract: 'posses' should be corrected to 'possess'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results on relations between properties of invariant σ-ideals and for recommending minor revision. The referee's description accurately reflects the paper's contributions. No specific major comments appear in the report.

Circularity Check

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No significant circularity detected

full rationale

The paper states direct proofs of implications between independently defined properties of invariant sigma-ideals (c-cc ideals are tall; Weaker Smital Property implies witnesses for non(I), ccc, and Fubini on Borel positive sets). No equations or steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The invariance assumption is explicit in the title and statements rather than smuggled. Derivations are self-contained against the stated definitions and Polish-space context.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works inside standard ZFC set theory with the usual definitions of Polish spaces and sigma-ideals; no free parameters, new entities, or ad-hoc axioms appear in the abstract.

axioms (1)

standard math ZFC set theory
All results are proved within classical set theory.

pith-pipeline@v0.9.0 · 5626 in / 1146 out tokens · 25684 ms · 2026-05-24T19:27:49.459714+00:00 · methodology

On some relations between properties of invariant σ-ideals in Polish spaces

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)