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arxiv: 1907.04344 · v1 · pith:QV6DRB43new · submitted 2019-07-09 · 🧮 math.GN

A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality

Angelo Bella , Santi Spadaro This is my paper

Pith reviewed 2026-05-24 23:52 UTC · model grok-4.3

classification 🧮 math.GN
keywords Hausdorff spacesGδ-modificationweak Lindelöf numbercardinal invariantsArhangel'skii theoremHajnal-Juhász inequalitygeneral topology
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The pith

A bound on the weak Lindelöf number of the Gδ-modification of Hausdorff spaces implies both Arhangel'skii's and Hajnal-Juhász's cardinal inequalities.

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

The paper establishes a single bound on the weak Lindelöf number of the Gδ-modification of any Hausdorff space. This bound yields the two classical results |X| ≤ 2^{L(X) χ(X)} and |X| ≤ 2^{c(X) χ(X)} as direct consequences. A sympathetic reader would care because these inequalities limit the size of a space in terms of its local character and covering numbers, two of the most studied relations in cardinal invariants of topology. The argument requires only the T2 axiom and thereby settles a question posed by Bell, Ginsburg and Woods.

Core claim

We present a bound for the weak Lindelöf number of the Gδ-modification of a Hausdorff space which implies various known cardinal inequalities, including |X| ≤ 2^{L(X) χ(X)} (Arhangel'skii) and |X| ≤ 2^{c(X) χ(X)} (Hajnal-Juhász). This solves a question that goes back to Bell, Ginsburg and Woods and is mentioned in Hodel's survey on Arhangel'skii's Theorem. In contrast to previous attempts we do not need any separation axiom beyond T2.

What carries the argument

The weak Lindelöf number wL of the Gδ-modification X_δ, which supplies the unifying upper bound from which the two classical size restrictions are derived.

If this is right

  • |X| ≤ 2^{L(X) χ(X)} holds for every Hausdorff space X
  • |X| ≤ 2^{c(X) χ(X)} holds for every Hausdorff space X
  • Other known cardinal inequalities in the literature follow from the same bound
  • The unification requires no separation axiom stronger than T2

Where Pith is reading between the lines

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

  • The same technique may produce bounds involving other modifications such as the regular or semiregular modification.
  • It would be natural to check whether the bound remains valid when L or c is replaced by other covering numbers such as the Lindelöf number itself.
  • Concrete examples in the class of compact Hausdorff spaces could be examined to test how close the new bound comes to equality.

Load-bearing premise

The bound on the weak Lindelöf number of the Gδ-modification holds for every Hausdorff space.

What would settle it

A Hausdorff space X in which wL(X_δ) strictly exceeds the bound given in the paper, so that neither classical inequality can be recovered from it.

read the original abstract

We present a bound for the weak Lindel\"of number of the $G_\delta$-modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\le 2^{L(X)\chi(X)}$ (Arhangel'skii) and $|X|\le 2^{c(X)\chi (X)}$ (Hajnal-Juhasz). This solves a question that goes back to Bell, Ginsburg and Woods and is mentioned in Hodel's survey on Arhangel'skii's Theorem. In contrast to previous attempts we do not need any separation axiom beyond $T_2$.

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.

Referee Report

0 major / 2 minor

Summary. The paper establishes a bound on the weak Lindelöf number wL(X_δ) of the Gδ-modification of any Hausdorff space X. This bound is shown to imply both Arhangel'skii's inequality |X| ≤ 2^{L(X) χ(X)} and the Hajnal-Juhász inequality |X| ≤ 2^{c(X) χ(X)}, thereby answering a question of Bell-Ginsburg-Woods. The argument relies only on the T2 separation axiom.

Significance. If the central bound on wL(X_δ) is correct, the result unifies two classical cardinal-function inequalities under a single statement that requires no separation axiom stronger than T2. The derivation directly yields the target inequalities via standard relations between the invariants and the Gδ-modification, without additional cardinal-arithmetic assumptions or post-hoc parameter choices.

minor comments (2)
  1. The notation for the Gδ-modification X_δ and the weak Lindelöf number wL should be defined explicitly in the first section rather than assumed from the abstract.
  2. A short remark clarifying how the new bound specializes to the two classical results (via the known inequalities wL(X_δ) ≤ L(X) and wL(X_δ) ≤ c(X)) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately captures the main contribution: a bound on wL(X_δ) that unifies Arhangel'skii's and Hajnal-Juhász's inequalities under the sole assumption of T₂.

Circularity Check

0 steps flagged

No significant circularity; new bound on wL(X_δ) is independently established and implies classical inequalities via standard relations

full rationale

The manuscript derives a bound on the weak Lindelöf number of the Gδ-modification of any Hausdorff space using only T2 separation to construct covers and dense sets. This bound is then shown to imply |X| ≤ 2^{L(X)χ(X)} and |X| ≤ 2^{c(X)χ(X)} through known relations between the invariants and the Gδ-modification. No step defines the new bound in terms of the target cardinalities, renames a known result, or relies on self-citation chains for its justification. The argument is self-contained against external benchmarks and does not reduce any claimed prediction to a fitted input or prior self-result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard properties of Hausdorff spaces and cardinal arithmetic but introduces no explicit free parameters, invented entities, or non-standard axioms.

axioms (2)
  • domain assumption The space is Hausdorff (T2).
    Explicitly stated as the only separation axiom required.
  • standard math Standard definitions and basic properties of the Lindelöf number, character, cellularity, and Gδ-modification hold.
    These are background facts from general topology used to state the bound and its implications.

pith-pipeline@v0.9.0 · 5647 in / 1360 out tokens · 19269 ms · 2026-05-24T23:52:09.608071+00:00 · methodology

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