Algebraic Topology Without Open Sets: A Net Approach to Homotopy Theory in Limit Spaces

Rodrigo Santos Monteiro

REVIEW 2 major objections 4 minor 36 references

Algebraic Topology Without Open Sets: A Net Approach to Homotopy Theory in Limit Spaces

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A groupoid Seifert–van Kampen theorem holds for limit spaces, with open covers replaced by convergence systems.

desk verdict The net-based development of limit-space homotopy is solid and the groupoid SvK theorem is new, but the proof of Theorem 4.0.1 has a concrete error in the colimit well-definedness that leaves the main claim unproven as written. read the letter →

arxiv 2412.11011 v2 pith:6HWJZBQX submitted 2024-12-15 math.AT math.GN

Rodrigo Santos Monteiro This is my paper · ORCID

classification math.ATmath.GN MSC 54A2055Q0518B40

keywords convergencespaceslimitnetsfundamentalgroupoidSeifert–vanKampentheoremhomotopytheorycontinuoussystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that homotopy theory can be developed for limit spaces using only net convergence, without open sets. It builds the fundamental groupoid of a limit space and proves that whenever a convergence system closed under finite intersections covers the space, the fundamental groupoid is the colimit of the fundamental groupoids of the pieces. If correct, this extends the classical groupoid Seifert–van Kampen theorem from topological spaces to all limit spaces, which include every topological space plus many non-topological convergence structures. The paper's concrete examples show this is a real extension: the fundamental groupoid of the real line with sequential convergence is discrete, unlike the usual one.

What carries the argument

The machinery has three parts. A limit space is a set with a convergence structure on nets that is centered, isotone, and stable under mixing of nets; a convergence system is a family $O$ of subsets such that every convergent net has a tail lying inside one member of $O$, playing the role an open cover plays classically. Paths and homotopies are defined through the exponential object provided by continuous convergence, making homotopy literally a path in the function space $C([0,1],X)$. The proof of Theorem 4.0.1 uses a Lebesgue-number subdivision: a path or homotopy is cut into small pieces, each contained in a single member of $O$, and the pieces are reassembled by the universal property of the colimit of the groupoids $\Pi(U)$.

What would settle it

Find a limit space $X$, a convergence system $O$ closed under finite intersections, and a path $\gamma:[0,1]\to X$ such that $\{\gamma^{-1}[U]:U\in O\}$ has no Lebesgue number; then the subdivision step in Theorem 4.0.1 cannot define $F([\gamma])$, so the claimed colimit equality would fail without further hypotheses.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.0.1: for a limit space $X$ and a convergence system $O$ of $X$ closed under finite intersections, the colimit of the functor sending each $U\in O$ to its fundamental groupoid $\Pi(U)$ is the fundamental groupoid $\Pi(X)$. Here paths are continuous functions from $[0,1]$ with its usual convergence, and homotopies are paths in the function space carrying continuous convergence, so no topology on $X$ is needed. The theorem is a groupoid version of Seifert–van Kampen for limit spaces, and it reduces to the classical topological statement when $O$ is an open cover. The paper also proves that $\Pi\colon \mathrm{LIM}\to\mathrm{GROUPOID}$ is a functor, that $\Pi(X\times Y)\cong \Pi(X)\times \Pi(Y)$, and that non-topological limit spaces can have fundamental groupoids different from those of their topological modifications.

Load-bearing premise

The load-bearing premise is that every path and homotopy in a limit space can be cut into finitely many small pieces, each lying entirely in one member of the convergence system, a Lebesgue-type property that the paper states as Lemma 4.0.1 without proof.

Editorial extensions

If this is right

For topological spaces, taking $O$ to be an open cover recovers the classical groupoid Seifert–van Kampen theorem, so the result is a strict generalization rather than a separate analogue.
The equality $\pi_1(X,x_0)=\Pi(X)[x_0,x_0]$ means fundamental groups of limit spaces can be computed by gluing local fundamental groupoids, with no path-connectedness or single-base-point assumption.
The functor $\Pi\colon\mathrm{LIM}\to\mathrm{GROUPOID}$ and the isomorphism $\Pi(X\times Y)\cong\Pi(X)\times\Pi(Y)$ give limit spaces a working toolkit for algebraic topology, including non-topological examples such as the lollipop space.
The example of the real line with sequential convergence shows that limit spaces carry genuinely new invariants: $\Pi(\langle\mathbb{R},\mathrm{Seq}\rangle)$ is discrete while the fundamental groupoid of the usual real line is not.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not pursued in the paper: if the theorem is right, $\Pi$ should be a 2-sheaf on the site whose covers are convergence systems closed under finite intersections, so the colimit statement would follow from a descent property; checking this would also show exactly where the finite-intersection hypothesis is used.
A likely testable strengthening is that the conclusion still holds when $O$ is replaced by its closure under finite intersections, since the proof only needs $U\cap V$ to make the cocone diagrams commute.
If the Lebesgue-type subdivision property fails for some limit space, the natural repair is to restrict to limit spaces whose convergence systems admit uniform subdivisions; the examples in the paper would still be covered.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the first three chapters are a clean, net-based introduction to convergence spaces and the fundamental groupoid, and the groupoid Seifert–van Kampen theorem for limit spaces is a genuinely new statement. But the proof of Theorem 4.0.1 has a concrete error in the well-definedness argument, and the main claim is not established as written.

What the paper does well: the net formalism is not new, but it is well executed and reads better than the filter-only treatments for someone with a geometric bent. The pasting lemma for limit spaces, the construction of Π(X), and the examples (discrete fundamental groupoid of (R, Seq), the lollipop) are all solid and clearly explained. The author is honest about the prior work of Rieser, Dossena, and Marroquín.

The soft spot is Theorem 4.0.1. Lemma 4.0.1 is asserted with 'the proof is the same as in the topological case'; that is a gap, though likely repairable. More seriously, step 2 of the proof contains an internal inconsistency. After subdividing the square and obtaining inclusions H[[0,t]×[0,s]]⊆U, H[[t,1]×[0,s]]⊆U′, H[[0,t]×[s,1]]⊆V, H[[t,1]×[s,1]]⊆V′, the path γ̃ defined by γ̃(r)=H(r,t) for r≤s and γ̃(r)=H(s,r) for r≥s is claimed to be a path in V. But for r∈[0,t], γ̃(r) lies in U; for r≥s, γ̃(r) lies in V′ (since s≥t). Also, t is the splitting time for γ, while γ′ needs its own splitting time. The later equalities F([γ̃0∗Γ̃])=F([γ]) depend on γ̃ being in V. This is not a harmless omission; the displayed construction cannot be correct. The theorem may still be true, and the bookkeeping may be repairable, but the current text does not prove it. A referee should ask for a corrected step 2 and a real proof of Lemma 4.0.1 before accepting the main claim.

Who is this for? Researchers in convergence spaces who want a net-friendly development and a candidate groupoid van Kampen theorem. I would send it to a serious referee, but with the expectation of major revision.

Referee Report

2 major / 4 minor

Summary. The manuscript develops a net-based foundation for convergence spaces and uses it to build homotopy theory in limit spaces. After introducing preconvergence, convergence, and limit spaces, it defines continuous convergence, establishes the existence of exponential objects, constructs the fundamental groupoid, and states a groupoid version of the Seifert-van Kampen theorem: for a limit space X and a convergence system O closed under finite intersections, the colimit of the functor U → Π(U) over U ∈ O is Π(X). The early chapters contain many carefully proved elementary results, and the paper includes instructive non-topological examples, such as the sequential convergence on R and the lollipop space. However, the proof of the central theorem, Theorem 4.0.1, contains a serious gap in the well-definedness argument for the colimit functor, and Lemma 4.0.1 is stated without proof.

Significance. If Theorem 4.0.1 can be repaired, the paper would provide a groupoid Seifert-van Kampen theorem for limit spaces, extending a classical topological result to spaces described only by net convergence. The net-based exposition is pleasant, and the examples showing that fundamental groupoids can change under non-topological limit structures are valuable. The compactness criterion in Theorem 2.4.1 and the careful treatment of continuous convergence are also strengths. Nevertheless, the central theorem is not fully supported as written, so the paper's main contribution currently rests on an unproved lemma and an erroneous subdivision argument.

major comments (2)

[Theorem 4.0.1, Step 2] The well-definedness argument for F on rel-homotopy classes is internally inconsistent. After choosing U, V, U', V' with H[[0,t]×[0,s]]⊆U, H[[t,1]×[0,s]]⊆U', H[[0,t]×[s,1]]⊆V, and H[[t,1]×[s,1]]⊆V', the text defines a path ~γ by ~γ(r)=H(r,t) for r≤s and ~γ(r)=H(s,r) for r≥s, and asserts that ~γ is a path in V. With the stated inclusions, for r∈[0,t] the point H(r,t) lies in U, and for r∈[t,s] it lies in U', so the assertion is false. Moreover, the splitting time t is taken for γ, while no splitting time for γ' is specified; the rectangle labels mix the two paths. Since the subsequent equalities F([~γ0*~Γ])=F([γ]) rely on this path and on the unproved claim that ~Γ is rel-homotopic to ⃗Γ*γ1, the construction of F on arrows is not established. A correct proof of well-definedness must be supplied.
[Lemma 4.0.1] Lemma 4.0.1 is stated without proof, with only 'the proof is the same as in the topological case'. This is not literally the classical Lebesgue lemma, because C is a convergence system and its members need not be open in the codomain. Theorem 4.0.1 invokes this lemma both for subdividing paths on [0,1] and for subdividing homotopies on [0,1]×[0,1], so the lemma is load-bearing. The manuscript should either prove that a convergence system on a compact metric domain has a Lebesgue number, or supply a precise reference that covers this setting.

minor comments (4)

[Section 3.3 / Theorem 4.0.1] Rel-homotopy is defined in Definition 3.3.1 as a map H:[0,1]→C([0,1],X), but Step 2 of Theorem 4.0.1 writes H:[0,1]×[0,1]→X. The exponential adjunction of Proposition 2.3.4 should be invoked explicitly, and the coordinate conventions should be fixed before the rectangle subdivision is discussed.
[Proposition 2.4.2(iii)] The last line of the proof of Proposition 2.4.2(iii) says 'inhL(A) ∩ inhL(B) ⊆ adhL(A ∩ B)', but the statement being proved is about inhL(A∩B); the final inclusion should be '⊆ inhL(A∩B)'.
[Example 2.1.6] Example 2.1.6 contains an unresolved placeholder 'φ↑#D, in the sense of the definition ??' and a misspelling 'satifsfayng'; these should be corrected.
[Section 3.3, groupoid composition] The composition law for Π(X) is written as ⟨[ρ],[γ]⟩ ↦ [γ∗ρ]. This is nonstandard unless composition is explicitly declared to be in diagrammatic order; please clarify the convention so that associativity and identities read correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central colimit Seifert–van Kampen proof is a direct universal-property argument, and the self-citations are not load-bearing.

full rationale

The derivation chain is not circular. The fundamental groupoid in Section 3.3 is defined directly from paths and rel-homotopies in a limit space, with associativity, identities, and inverses verified inside the paper using the Pasting Lemma (Lemma 3.1.1), which is proven there. Homotopy is defined via the exponential object of continuous convergence, whose main properties (Propositions 2.3.2–2.3.4) are proved in the text. Theorem 4.0.1 is a genuine universal-property proof: given any cocone {F_U : Π(U) → G}, the functor F is defined on objects by F(x)=F_U(x) (well-defined by cocone commutativity) and on arrows by subdividing a path using the convergence system and composing the corresponding F_U-values. The proof then attempts to show this definition is independent of the subdivision and of rel-homotopy. None of these steps fits a parameter, renames a known result, or invokes a conclusion as its own hypothesis. The self-citations in the paper—the introduction's statement that the work extends the author's previous paper [21] and Remark 3.1.1's note that a more general Pasting Lemma appears in [21]—are not used to prove Theorem 4.0.1 or any other load-bearing result. Lemma 4.0.1's proof is omitted with a reference to the topological case in the external textbook [14]; this is an incompleteness in the written proof, not a circular reliance on an unverified self-citation. The sketched Step 2 of Theorem 4.0.1 may contain a concrete rectangle-labeling mistake, as the path ~γ claimed to lie in V includes a segment lying in U according to the displayed inclusions; however, a flawed or sketchy argument is a correctness concern, not a circularity. There are no fitted inputs, no statistical forcing, and no derived quantity that equals its input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles, forces, dimensions, or empirical constants. Its free parameters are empty. The axioms are standard set-theoretic and topological principles plus two assumptions specific to the main theorem: the convergence-system Lebesgue lemma and the well-definedness of the colimit functor. These two assumptions are precisely where the proof is incomplete.

assumptions (4)

standard math Every proper filter is induced by a net, and net convergence and filter convergence are interchangeable.
Used throughout the paper to convert between net and filter formulations, for example in Proposition 1.2.4 and in the definition of preconvergence.
standard math The Axiom of Choice is accepted to select points from nonempty intersections and to build nets indexed by neighborhood systems.
Invoked in Proposition 1.2.1 and Proposition 2.4.1 to construct nets from filter-based adherence arguments.
ad hoc to paper Lebesgue's covering lemma extends to convergence systems on compact metric spaces.
Lemma 4.0.1 asserts this extension without proof. It is load-bearing for the subdivision of paths and homotopies in the main Seifert-van Kampen theorem.
ad hoc to paper A convergence system closed under finite intersections generates a colimit diagram whose cocone to the fundamental groupoid is universal.
This is effectively the content of Theorem 4.0.1. The proof assumes that the functor F on arrows is well-defined, but the rel-homotopy case is only sketched.

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Pith. "Pith review of Algebraic Topology Without Open Sets: A Net Approach to Homotopy Theory in Limit Spaces." pith.science (2026). https://pith.science/paper/6HWJZBQX

@misc{pith2026241211011,
  author       = {Pith},
  title        = {Pith review of: Algebraic Topology Without Open Sets: A Net Approach to Homotopy Theory in Limit Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HWJZBQX}},
  note         = {Machine review of arXiv:2412.11011}
}

read the original abstract

Convergence spaces are a generalization of topological spaces. The category of convergence spaces is well-suited for Algebraic Topology, one of the reasons is the existence of exponential objects provided by continuous convergence. In this work, we use a net-theoretic approach to convergence spaces. The goal is to simplify the description of continuous convergence and apply it to problems related to homotopy theory. We present methods to develop the basis of homotopy theory in limit spaces, define the fundamental groupoid, and prove the groupoid version of the Seifert-van Kampen Theorem for limit spaces.

Figures

Figures reproduced from arXiv: 2412.11011 by the authors.

**Figure 1.1.** Suppose that A, B, C and D are elements of a filter. Condition 1 in Definition 1.1.1 allows us to find increasingly smaller sets the filter, which gives us an idea of approximation or convergence. Example 1.1.1. Let X be a topological space. For each x ∈ X, the family Nx of neighborhoods of x, is a filter. Indeed, if A, B ∈ Nx there are open sets U, V ⊆ X such that x ∈ U ∩ V , U ⊆ A and V ⊆ B. Notice that U ∩ V ⊆ A … view at source ↗

**Figure 1.2.** There is a moment from which all terms of the net are within the neighbor [PITH_FULL_IMAGE:figures/full_fig_p018_1_2.png] view at source ↗

**Figure 1.3.** The partition in blue is better than the in red [PITH_FULL_IMAGE:figures/full_fig_p018_1_3.png] view at source ↗

Figures from the paper (15 more)

**Figure 1.4.** Figure 1.4: A tag in green of the partition in blue Let X be a real vector space, and f : [a, b] → X and α : [a, b] → R be functions. For a tagged partition ⟨P, T⟩ ∈ P ∗ [a, b], we associated the Riemann-Stieltjes sum of f and α over ⟨P, T⟩ X ⟨P,T⟩ f, α = Xn i=1 f(ti) · (α(pi) −…

**Figure 1.5.** Figure 1.5: A subnet(green) of a net(red) converging to the same point [PITH_FULL_IMAGE:figures/full_fig_p021_1_5.png]

**Figure 1.6.** Figure 1.6: Two nets (pink and red) converging to the same point (blue) and their mixing [PITH_FULL_IMAGE:figures/full_fig_p023_1_6.png]

**Figure 2.1.** Figure 2.1: Consider a net (in pink) that starts at x0 and moves in a counterclockwise direction, and after the first cycle, it only takes the values x0, x1 or x2. In this case, the convergence is witnessed by the points x0, x1 and x2. ◀ Example 2.1.5 (Adapted from [27]). In the…

**Figure 2.2.** Figure 2.2: The lollipop X Let S = L \ {p} and D ⊆ S be a dense countable set with respect to the usual topology of R 2 . For a net φ ∈ NETS(X) and a point x ∈ X, we define a preconvergence λ such as: 1. If x ̸= p, then φ →λ x if and only if φ →R2 x, 2. If x = p, then φ →λ x if …

**Figure 2.3.** Figure 2.3: A pictorial description of this argument [PITH_FULL_IMAGE:figures/full_fig_p035_2_3.png]

**Figure 2.4.** Figure 2.4: An illustration of how continuous convergence happens. [PITH_FULL_IMAGE:figures/full_fig_p039_2_4.png]

**Figure 2.5.** Figure 2.5: There is some element of the convergence system that contains a tail of the [PITH_FULL_IMAGE:figures/full_fig_p045_2_5.png]

**Figure 3.1.** Figure 3.1: Intuitively, the idea of path gluing involves finding a way to traverse two [PITH_FULL_IMAGE:figures/full_fig_p049_3_1.png]

**Figure 3.2.** Figure 3.2: The idea of the last proof was to collapse to the limit point all the point in [PITH_FULL_IMAGE:figures/full_fig_p050_3_2.png]

**Figure 3.3.** Figure 3.3: The diference between rel-homotopy and homotopy: On the left, we have a [PITH_FULL_IMAGE:figures/full_fig_p054_3_3.png]

**Figure 3.4.** Figure 3.4: An illustration of the presented argument. [PITH_FULL_IMAGE:figures/full_fig_p058_3_4.png]

**Figure 3.5.** Figure 3.5: To obtain this space, the idea is to attach a lollipop at each point of the [PITH_FULL_IMAGE:figures/full_fig_p059_3_5.png]

**Figure 4.1.** Figure 4.1: An illustration of how we can see γ as a concatenation. 1. Let us show that F([γ]) does not depend on the subdivision and the choice of the sets Ui . For this, it is enough to prove the case γ = γ1 ∗γ2, where the images of γ, γ1 and γ2 are contained is U0, U1 and U2,…

**Figure 4.2.** Figure 4.2: The idea is move the path γ to γ ′ by homotopies relative to the endpoints through the squares. Consider γ = γ0 ∗ γ1 and γ ′ = γ ′ 0 ∗ γ ′ 1 where the image of γ0 is contained in U, the image of γ1 is contained in V , the image of γ ′ 0 is contained in U ′ and the im…

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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