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arxiv: 2604.26730 · v2 · pith:M7WQPA6Knew · submitted 2026-04-29 · 🧮 math.GR · math.GN

On the existence and properties of Alexandroff paratopological groups

Pedro J. Chocano , Tayomara Borsich This is my paper

Pith reviewed 2026-05-19 17:21 UTC · model grok-4.3

classification 🧮 math.GR math.GN
keywords Alexandroff topologyparatopological grouptopological groupfeebly bounded setT0 spacenon-compact space
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The pith

No non-discrete Alexandroff topology can turn a group into a topological group.

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

The paper establishes that Alexandroff topologies, where arbitrary intersections of open sets remain open, cannot support non-discrete topological groups because the continuity of operations forces every point to be isolated. This negative result shifts focus to Alexandroff paratopological groups, which relax the continuity requirements. In this setting the authors construct explicit non-compact T0 examples and use them to prove that products of feebly bounded sets are feebly bounded and that the square of a feebly bounded set is also feebly bounded, thereby answering two open questions.

Core claim

No non-discrete Alexandroff topology can make the group operations continuous. Consequently the authors develop the theory of Alexandroff paratopological groups, prove several fundamental properties, exhibit concrete non-compact T0 instances, and apply the framework to show that the product of any two feebly bounded subsets is feebly bounded while the square of a feebly bounded subset remains feebly bounded.

What carries the argument

The Alexandroff property of a topology, under which the intersection of any family of open sets is open, which creates an obstruction to continuity of group multiplication and inversion unless the topology is discrete.

If this is right

  • Alexandroff topological groups, if they exist, must be discrete.
  • Non-compact T0 Alexandroff paratopological groups exist.
  • The product of feebly bounded subsets in these groups is feebly bounded.
  • If B is a feebly bounded subset then B squared is also feebly bounded.

Where Pith is reading between the lines

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

  • Similar intersection-closed topologies may face the same discreteness barrier in topological groups.
  • The constructed examples could serve as test cases for other open problems in paratopological group theory.
  • Extending the negative result to semigroups or other structures with continuous operations might be possible.

Load-bearing premise

That the topology must be Alexandroff, so that arbitrary intersections of open sets stay open, which is the key property used to derive that continuity implies discreteness.

What would settle it

A specific non-discrete group equipped with an Alexandroff topology in which both multiplication and inversion maps are continuous would disprove the non-existence result.

Figures

Figures reproduced from arXiv: 2604.26730 by Pedro J. Chocano, Tayomara Borsich.

Figure 1
Figure 1. Figure 1: Hasse diagram of Z ⊕ Z from Example 4.7. Theorem 4.8. Let X be a non-compact Alexandroff paratopological group. Then a point x ∈ X is a beat point if and only if X is isomorphic to the Alexandroff paratopological group (Z, ≤). In particular, X is contractible. Proof. We only prove the nontrivial implication. Without loss of generality, we assume that x is an up beat point. By Corollary 4.6, every point x ∈… view at source ↗
Figure 2
Figure 2. Figure 2: Hasse diagram of X from Example 5.2. Definition 5.3. Let X be a non-compact Alexandroff paratopological group. We define the upper ball of minimal radius of x ∈ X, denoted by r(x), as r(x) = {y ∈ X | x ≺ y}. Moreover, we define the radius of X to be |r(1)|, and we denote it by r(X). Clearly, the radius is a topological invariant and |r(x)| = |r(1)| for every x ∈ X by Theorem 4.4. However, this invariant is… view at source ↗
read the original abstract

We study groups endowed with Alexandroff topologies and show that no non-discrete Alexandroff topology can turn a group into a topological group. This settles negatively the basic existence problem for Alexandroff topological groups. Motivated by this obstruction, we turn to the broader setting of Alexandroff paratopological groups. We establish several fundamental properties of these spaces and provide explicit non-compact $T_0$ examples, showing that the Alexandroff framework is rich enough to capture nontrivial paratopological phenomena. As applications, we address two classical open questions concerning feebly bounded subsets in paratopological groups, proving that non-compact Alexandroff paratopological groups offer a positive solution both for products of feebly bounded sets and for the feebly boundedness of $B^2$ when $B$ is a feebly bounded subset.

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 / 2 minor

Summary. The manuscript proves that no non-discrete Alexandroff topology can make a group into a topological group, negatively settling the existence question for Alexandroff topological groups. It then develops the theory of Alexandroff paratopological groups by establishing fundamental properties, constructing explicit non-compact T0 examples, and applying these constructions to resolve two open questions on feebly bounded subsets: the product of feebly bounded sets is feebly bounded, and B² is feebly bounded whenever B is feebly bounded.

Significance. The negative result on topological groups is definitive and follows directly from the Alexandroff intersection property together with joint continuity of multiplication and continuity of inversion. The positive applications to feebly bounded sets provide concrete counterexamples to prior conjectures in the non-compact setting and demonstrate that the Alexandroff framework is sufficiently rich to capture nontrivial paratopological phenomena while preserving T0 separation. Explicit constructions and parameter-free derivations strengthen the contribution.

major comments (1)
  1. [§2] §2, Theorem 2.3: the argument that the Alexandroff property forces every singleton to be open once a neighborhood basis at the identity is fixed appears load-bearing for the discreteness claim; please confirm that the proof does not inadvertently assume local compactness or any separation axiom beyond T0 when handling the inversion map.
minor comments (2)
  1. [Introduction] The statement of the two open questions on feebly bounded sets in the introduction would benefit from explicit citations to the original sources rather than paraphrases.
  2. [§4] In the construction of the non-compact T0 examples (likely §4), the verification that the topology is Alexandroff could be made more explicit by listing the subbasis elements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for minor revision. The negative result on Alexandroff topological groups and the applications to feebly bounded sets are correctly summarized. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [§2] §2, Theorem 2.3: the argument that the Alexandroff property forces every singleton to be open once a neighborhood basis at the identity is fixed appears load-bearing for the discreteness claim; please confirm that the proof does not inadvertently assume local compactness or any separation axiom beyond T0 when handling the inversion map.

    Authors: We thank the referee for this observation. The proof of Theorem 2.3 relies only on the Alexandroff intersection property and the definition of a topological group (joint continuity of multiplication together with continuity of inversion). After fixing a neighborhood basis at the identity e, the Alexandroff property yields a smallest open neighborhood U of e. Continuity of inversion maps U to a neighborhood of e; minimality of U then forces U = U^{-1}. Joint continuity of multiplication at (e,e) combined with the T0 axiom (used solely to separate e from other group elements) implies that U must be the singleton {e}, rendering the topology discrete. The argument is strictly local at the identity and invokes neither local compactness nor any separation axiom stronger than T0. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central result—that no non-discrete Alexandroff topology yields a topological group—follows deductively from the definition of Alexandroff spaces (arbitrary intersections of open sets remain open) together with the joint continuity of multiplication and continuity of inversion in a topological group. This forces every singleton to be open, implying discreteness. No parameters are fitted, no self-citations serve as load-bearing uniqueness theorems, and no ansatz or renaming of prior results is invoked to establish the obstruction. The subsequent constructions of Alexandroff paratopological groups and applications to feebly bounded sets are likewise built from explicit examples and standard properties without reducing to the input claims by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard domain assumptions from general topology and topological algebra; no free parameters or new entities are introduced based on the abstract description.

axioms (2)
  • domain assumption Alexandroff topologies are those in which the intersection of any family of open sets is open.
    This is the core definition invoked for the study of these spaces as per the abstract.
  • domain assumption A paratopological group is a group with a topology making the multiplication map continuous.
    Standard definition used to extend from the topological group case.

pith-pipeline@v0.9.0 · 5680 in / 1339 out tokens · 75474 ms · 2026-05-19T17:21:28.463041+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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