arxiv: 2605.11397 · v1 · submitted 2026-05-12 · 🧮 math.CO · math.GN· math.LO

Recognition: no theorem link

On minimal collections of sequences for testing continuity

Gyuhyun Lim

Pith reviewed 2026-05-13 02:17 UTC · model grok-4.3

classification 🧮 math.CO math.GNmath.LO

keywords test setscontinuity detectionconvergent sequencesposetminimal elementssequential fandiscontinuities

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The pith

Minimal test sets of convergent sequences exist for detecting discontinuities at a point under natural hypotheses.

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

The paper defines test sets as subfamilies of sequences converging to a point P that still suffice to detect every discontinuity of any real-valued function at P. These test sets are partially ordered by inclusion. Under natural hypotheses at P, the poset is proven to contain minimal elements. The authors also examine maximal chains in the poset, finding that some possess a least element and others do not. A concrete construction on the sequential fan shows that a minimal test set can have strictly smaller cardinality than the collection of all convergent sequences.

Core claim

Under natural hypotheses at P, the poset of test sets has a minimal element. Maximal chains in this poset may or may not have a least element. On the sequential fan, the construction produces a minimal test set whose cardinality is strictly smaller than that of the full family of convergent sequences.

What carries the argument

The poset of test sets: subfamilies of sequences converging to P that detect every discontinuity at P, ordered by inclusion.

Load-bearing premise

The unspecified natural hypotheses at the point P are satisfied.

What would settle it

A point P satisfying the natural hypotheses for which no minimal test set exists would falsify the main existence claim.

Figures

Figures reproduced from arXiv: 2605.11397 by Gyuhyun Lim.

**Figure 1.** Figure 1: A schematic picture of the two chain behaviors established in this paper. On the left, a bad maximal chain does not induce a minimal test set. On the right, a good maximal chain has intersection equal to a minimal test set. Outline of the Paper. In Section 2, we introduce the common sequential language used throughout the paper and formalize the notions of witness families, test sets, and maximal chains. … view at source ↗

**Figure 2.** Figure 2: Prefix-fixing families An(a) shrink to {a}, while removing a forces T n Bn(a) = ∅. Proof. The inclusion Bn+1 ⊆ Bn is immediate from An+1 ⊆ An. For nonemptiness, fix n. By Lemma 2.9, the space X has some point different from an+1; choose x ∈ X with x ̸= an+1. Define a sequence T = (Tk) by Tk = ( ak, k ̸= n + 1, x, k = n + 1. Then T agrees with a for all sufficiently large k, so by Lemma 3.1(i) we have T ∈ S… view at source ↗

**Figure 3.** Figure 3: The countable sequential fan Sω. Each spoke is Bn = {xn,m : m ∈ N} and only five labeled spokes are shown explicitly; the fan has countably many spokes. The red segment near P represents the tail Uf ∩Bn of a basic neighborhood Uf . The canonical sequence Tn = (xn,k)k∈N along each spoke is an element of the minimal test set. The space Sω satisfies the standing framework. Indeed, it is T1 because every xn,m … view at source ↗

read the original abstract

We study test sets: subfamilies of sequences converging to a point P that still suffice to detect every discontinuity of real-valued functions at P. Ordered by inclusion, these test sets form a poset. Under natural hypotheses at P, we prove that this poset has a minimal element. We also analyze its maximal chains, showing that some have a least element, while others do not. Finally, on the sequential fan we give a concrete realization in which the minimal test set produced by our construction has strictly smaller cardinality than the full family of convergent sequences.

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 frames continuity test sets as a poset, proves minimal elements exist under natural hypotheses, and shows a strictly smaller minimal set on the sequential fan.

read the letter

The main points are that test sets — subfamilies of sequences to P that still catch every discontinuity of real functions at P — form a poset under inclusion, that this poset has minimal elements under natural hypotheses at P, that maximal chains behave in mixed ways, and that on the sequential fan the construction yields a minimal test set with strictly smaller cardinality than the full family of convergent sequences. That last comparison is the most concrete part and shows the idea can produce savings in a standard example space. The poset organization itself is a clean way to talk about minimality for sequential testing, and the existence claim plus the chain analysis give a structured picture rather than isolated examples. The work stays focused on existence and comparison without inflating the scope. The sequential fan example is a reasonable choice because it is a space where sequences matter but the topology is not first-countable, so a smaller test family is worth exhibiting. The soft spot is the unspecified natural hypotheses. The existence proof almost certainly applies Zorn or a similar argument once every chain has a lower bound that remains a test set, but without the exact list it is hard to tell how often those hypotheses hold or whether they are close to the conclusion itself. The chain analysis also needs the details to see whether the distinction between chains with and without least elements adds real content. No circularity or fitting issues appear from what is described. This is for people working in general topology or set-theoretic topology who already care about sequential properties and minimal families. A reader who studies the sequential fan or poset methods in topology will get the most out of the cardinality example and the organizational frame. It is narrow enough that it will not interest a wide audience, but the concrete construction is worth checking. I would send it to referees rather than desk-reject; the idea is straightforward and the example is specific enough that experts can verify the hypotheses and the construction in one pass.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the poset of test sets (subfamilies of sequences converging to a point P that suffice to detect all discontinuities of real-valued functions at P), ordered by inclusion. Under natural hypotheses at P, it claims to prove that this poset has minimal elements, analyzes its maximal chains (showing some have a least element while others do not), and gives a concrete construction on the sequential fan in which a minimal test set has strictly smaller cardinality than the full family of convergent sequences.

Significance. If the central claims hold with explicit hypotheses, the work would offer a new order-theoretic perspective on minimal sufficient families for continuity testing in sequential spaces, potentially aiding both theoretical topology and practical discontinuity detection. The sequential-fan example provides a falsifiable, concrete cardinality reduction that strengthens the contribution; the maximal-chain analysis adds structural insight.

major comments (1)

[Abstract] Abstract (and presumably the statement of the main theorem): The 'natural hypotheses at P' are never defined or listed. The existence proof for minimal elements requires these hypotheses to guarantee that every chain of test sets has a lower bound that remains a test set (so that Zorn's lemma or an equivalent minimality argument applies). Without an explicit list, it is impossible to verify whether the hypotheses hold at points of interest or merely restate the conclusion, rendering the central claim unverifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (and presumably the statement of the main theorem): The 'natural hypotheses at P' are never defined or listed. The existence proof for minimal elements requires these hypotheses to guarantee that every chain of test sets has a lower bound that remains a test set (so that Zorn's lemma or an equivalent minimality argument applies). Without an explicit list, it is impossible to verify whether the hypotheses hold at points of interest or merely restate the conclusion, rendering the central claim unverifiable.

Authors: We agree that the abstract does not contain an explicit enumerated list of the hypotheses. The manuscript introduces the relevant conditions on the point P in the introduction and develops them fully in Section 2 (including the requirement that intersections of chains of test sets remain test sets, which licenses the application of Zorn's lemma). To improve immediate verifiability, we will revise the abstract and the statement of the main theorem to include a concise, explicit list of these hypotheses. The proof itself will remain unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity; direct existence proof in order theory with no reductions to inputs or self-citations.

full rationale

The paper proves existence of a minimal element in the poset of test sets (subfamilies of sequences to P sufficient to detect discontinuities) under natural hypotheses at P, analyzes maximal chains, and gives a concrete smaller-cardinality example on the sequential fan. No equations, fitted parameters, or self-citations appear that reduce any claimed prediction or minimal element to a definition or prior result by construction. The argument is a standard order-theoretic existence proof (likely via Zorn or chain bounds) that remains independent of its inputs; the hypotheses are external to the derivation rather than smuggled in via self-reference. This matches the default expectation of a non-circular math paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions of sequence convergence, discontinuity of real-valued functions, and the partial order of inclusion on families of sequences; no free parameters or invented entities are introduced.

axioms (2)

domain assumption Natural hypotheses at point P that guarantee the poset of test sets has minimal elements
The proof of existence of a minimal test set is stated to hold only under these unspecified hypotheses at P.
standard math Standard axioms of topology and order theory for sequences converging to a point
The entire construction of test sets and the poset relies on the usual notions of convergence and inclusion in topological spaces.

pith-pipeline@v0.9.0 · 5375 in / 1425 out tokens · 58279 ms · 2026-05-13T02:17:35.973380+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

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R. Engelking, General Topology, 2nd ed., Sigma Series in Pure Mathematics, Vol. 6, Heldermann Verlag, Berlin, 1989

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[2]

S. P. Franklin, Spaces in which sequences suffice, Fundamenta Mathematicae 57 (1965), 107--115

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[3]

S. P. Franklin and B. V. Smith Thomas, On the metrizability of \(k_ \)-spaces, Pacific Journal of Mathematics 72 (1977), no. 2, 399--402

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[4]

Hru s \' a k, Almost disjoint families and topology, in Recent Progress in General Topology III (K

M. Hru s \' a k, Almost disjoint families and topology, in Recent Progress in General Topology III (K. P. Hart, J. van Mill, P. Simon, eds.), Atlantis Press, Paris, 2014, 601--638

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[5]

Jech, Set Theory: The Third Millennium Edition, revised and expanded, Springer Monographs in Mathematics, Springer, Berlin--Heidelberg, 2003

T. Jech, Set Theory: The Third Millennium Edition, revised and expanded, Springer Monographs in Mathematics, Springer, Berlin--Heidelberg, 2003

work page 2003