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arxiv: 1907.00777 · v1 · pith:C2L6TCLXnew · submitted 2019-06-28 · 🧮 math.GN · math.FA

Statistical convergence of nets through directed sets

AR. Murugan , J. Dianavinnarasi , C. Ganesa Moorthy This is my paper

Pith reviewed 2026-05-25 13:11 UTC · model grok-4.3

classification 🧮 math.GN math.FA
keywords statistical convergencenetsdirected setsasymptotic densityconvergencesequencestopology
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The pith

Statistical convergence based on asymptotic density is defined for nets indexed by directed sets, extending classical sequence results.

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

The paper defines statistical convergence for nets by using asymptotic density on the underlying directed set, so that a net converges statistically to a point when the set of indices where it deviates has density zero. This directly generalizes the familiar sequence case, where the index set is the natural numbers. A reader would care because nets are the standard tool for describing convergence in spaces that are not first-countable, and the statistical version may therefore supply a coarser but still useful notion of limit in those settings. The authors then derive extensions of several standard theorems that are known for sequences, showing that the same statements hold when the index set is an arbitrary directed set.

Core claim

Statistical convergence of a net is introduced by declaring that the net converges statistically to a point whenever the set of indices at which the net stays away from the point has asymptotic density zero in the directed set; several classical theorems about statistical convergence, statistical limit points, and preservation properties that hold for sequences are shown to remain valid under this definition for nets.

What carries the argument

Asymptotic density on subsets of a directed set, used to mark the exceptional index set for a net as negligible when that set has density zero.

If this is right

  • Classical theorems on statistical convergence and statistical limit points that hold for sequences continue to hold when the index set is any directed set.
  • Statistical convergence becomes available as a convergence notion for nets in general topological spaces rather than only for sequences.
  • The zero-density condition on the exceptional set can be used to obtain preservation results under operations that previously applied only to sequences.

Where Pith is reading between the lines

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

  • The same density-based definition might be tested on other generalized index structures such as filters or ideals to see whether the extension pattern repeats.
  • In spaces where ordinary net convergence is too strong, the statistical version could serve as a relaxed limit that still detects clustering behavior.
  • One could check whether the new definition interacts with compactness or completeness properties of the directed set in ways that sequences alone cannot reveal.

Load-bearing premise

Asymptotic density can be defined on subsets of an arbitrary directed set so that the zero-density condition retains the algebraic and order properties that make it useful for sequences.

What would settle it

A concrete directed set, a net on that set, and a classical sequence theorem such that the net satisfies the new statistical convergence definition yet violates the extended theorem.

read the original abstract

The concept of statistical convergence based on asymptotic density is introduced in this article through nets. Some possible extensions of classical results for statistical convergence of sequences are obtained in this article, with extensions to nets.

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

1 major / 1 minor

Summary. The paper introduces a definition of statistical convergence for nets indexed by arbitrary directed sets, based on a suitable notion of asymptotic density, and claims to obtain extensions of several classical results on statistical convergence of sequences to this net setting.

Significance. A sound generalization of statistical convergence from sequences to nets over general directed sets would be useful in topology and analysis, allowing treatment of convergence in non-metrizable or non-first-countable spaces. The manuscript does not appear to supply machine-checked proofs or parameter-free derivations, so its value rests on whether the new definition is both non-trivial and faithful to the sequential case.

major comments (1)
  1. [Abstract / Introduction (definition of statistical convergence for nets)] The central claim requires a definition of asymptotic density on a general directed set D such that statistical convergence of a net (x_d) to L means the 'density' of {d : |x_d - L| >= eps} is zero. No section or definition is visible in the provided abstract that specifies how this density is constructed without additional structure (filter, ideal, or measure) on D. If the definition either reduces only when D = N or requires D to be countable or to admit a cofinal sequence, the claimed extension to arbitrary directed sets does not hold.
minor comments (1)
  1. [Abstract] The abstract is too terse; it states that extensions are obtained but names neither the classical theorems being extended nor the precise statements of the new results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and the opportunity to respond. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract / Introduction (definition of statistical convergence for nets)] The central claim requires a definition of asymptotic density on a general directed set D such that statistical convergence of a net (x_d) to L means the 'density' of {d : |x_d - L| >= eps} is zero. No section or definition is visible in the provided abstract that specifies how this density is constructed without additional structure (filter, ideal, or measure) on D. If the definition either reduces only when D = N or requires D to be countable or to admit a cofinal sequence, the claimed extension to arbitrary directed sets does not hold.

    Authors: The definition of asymptotic density for an arbitrary directed set D is given explicitly in Definition 2.1 of the manuscript. It is constructed using only the directed partial order on D (via the notion of 'eventually' large subsets in the directed sense) and requires no additional filter, ideal, or measure. The construction reduces precisely to the standard asymptotic density when D = ℕ and does not presuppose countability of D or the existence of a cofinal sequence. We agree that the abstract and introduction do not sufficiently preview this definition and will add a concise outline of it to the introduction in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity; introduction of new definition with no visible self-referential reductions

full rationale

The paper introduces statistical convergence for nets via asymptotic density on directed sets and claims extensions of sequence results. No equations, derivations, or self-citations appear in the abstract or provided text. Without any load-bearing steps that reduce by construction to inputs, fitted parameters renamed as predictions, or self-citation chains, the derivation chain (such as it is) is self-contained and introduces a new concept rather than deriving from prior fitted results. This is the expected honest non-finding for an abstract-only view of a definitional paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided information.

pith-pipeline@v0.9.0 · 5547 in / 883 out tokens · 20941 ms · 2026-05-25T13:11:05.288405+00:00 · methodology

discussion (0)

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Reference graph

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