arxiv: 2604.26730 · v1 · submitted 2026-04-29 · 🧮 math.GR · math.GN

Recognition: unknown

On the existence and properties of Alexandroff paratopological groups

Pedro J. Chocano, Tayomara Borsich

Pith reviewed 2026-05-07 11:48 UTC · model grok-4.3

classification 🧮 math.GR math.GN

keywords Alexandroff topologyparatopological groupstopological groupsfeebly bounded setsT0 spacesgroup operationsnon-compact spaces

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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 proves that equipping a group with a non-discrete Alexandroff topology prevents the group operations from being continuous. An Alexandroff topology requires that arbitrary intersections of open sets are open. This obstruction leads the authors to study Alexandroff paratopological groups instead, where only multiplication needs to be continuous. They construct explicit non-compact T0 examples and use them to answer two open questions about feebly bounded subsets in paratopological groups.

Core claim

The paper establishes that there do not exist any non-discrete Alexandroff topological groups. Motivated by this, it introduces Alexandroff paratopological groups, derives their fundamental properties, and gives concrete non-compact T0 examples. These examples then provide positive answers to classical open problems on whether products of feebly bounded sets are feebly bounded and whether the square of a feebly bounded set is feebly bounded in paratopological groups.

What carries the argument

The Alexandroff property, which demands that the intersection of any family of open sets remains open, when imposed on the topology of a group or paratopological group.

Load-bearing premise

The topology on the group must have the property that arbitrary intersections of open sets are open, while the operations are required to be continuous.

What would settle it

A counterexample would be a group equipped with a non-discrete Alexandroff topology making both multiplication and inversion continuous, or an Alexandroff paratopological group where the product of two feebly bounded sets is not feebly bounded.

Figures

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

**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: 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.

Desk Editor's Note private letter to a colleague

The paper's headline non-existence result for non-discrete Alexandroff topological groups is false, as the indiscrete topology is a counterexample.

read the letter

The central claim that no non-discrete Alexandroff topology can make a group into a topological group does not hold. The indiscrete topology on any group with more than one element has open sets empty and the whole group, so arbitrary intersections stay open and it is Alexandroff. Multiplication and inversion are continuous because their preimages of the whole group are the full product space and preimages of empty are empty. The abstract states this result without any separation axiom, though T0 appears only for the later paratopological constructions. That gap makes the non-existence theorem incorrect on its face.

Referee Report

1 major / 0 minor

Summary. The paper proves that no non-discrete Alexandroff topology on a group makes the group operations continuous, negatively settling the existence question for Alexandroff topological groups. It then develops the theory of Alexandroff paratopological groups, establishes basic properties, constructs explicit non-compact T0 examples, and applies these constructions to resolve two open questions on feebly bounded subsets (products of feebly bounded sets remain feebly bounded, and B² is feebly bounded whenever B is).

Significance. If the non-existence result holds under the intended hypotheses, it would cleanly resolve a basic existence problem for Alexandroff topological groups. The explicit constructions of non-compact T0 Alexandroff paratopological groups and the positive solutions to the two open questions on feebly bounded sets would supply concrete examples and new techniques useful to researchers working on paratopological groups and boundedness properties.

major comments (1)

[Abstract] Abstract: The central claim that 'no non-discrete Alexandroff topology can turn a group into a topological group' is false as stated. The indiscrete topology on any group G with |G|>1 is Alexandroff (arbitrary intersections of open sets remain open), non-discrete, and yields a topological group because the preimage of the only nonempty open set G under multiplication or inversion is the entire domain. The paper invokes T0 only for the later paratopological examples, indicating that a separation axiom is likely intended but omitted from the topological-group non-existence theorem; this omission is load-bearing for the headline result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to clarify the separation axioms in the statement of our non-existence result. We agree that the abstract as written is imprecise and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that 'no non-discrete Alexandroff topology can turn a group into a topological group' is false as stated. The indiscrete topology on any group G with |G|>1 is Alexandroff (arbitrary intersections of open sets remain open), non-discrete, and yields a topological group because the preimage of the only nonempty open set G under multiplication or inversion is the entire domain. The paper invokes T0 only for the later paratopological examples, indicating that a separation axiom is likely intended but omitted from the topological-group non-existence theorem; this omission is load-bearing for the headline result.

Authors: We appreciate the referee's observation. The indiscrete topology is indeed a non-discrete Alexandroff topological group. Our proof in the body of the paper actually establishes the non-existence result under the additional T0 separation axiom (which is used to obtain the contradiction with continuity of the group operations in the non-discrete case). This hypothesis was inadvertently omitted from the abstract and the formal statement of the theorem, although it is consistent with the T0 examples constructed for the paratopological case. We will revise the abstract, the introduction, and the theorem statement to explicitly include the T0 assumption, rendering the claim accurate. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained.

full rationale

The paper establishes non-existence of non-discrete Alexandroff topological groups via direct analysis of continuity of group operations under the Alexandroff intersection property, then shifts to explicit constructions of T0 Alexandroff paratopological groups and applications to feebly bounded sets. No step reduces by definition to its own output, no parameters are fitted and relabeled as predictions, and no load-bearing premise rests on a self-citation chain or imported uniqueness theorem. The central non-existence argument relies on standard topological and algebraic properties without circular reduction, consistent with the reader's assessment of only minor or absent circular elements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies exclusively on standard axioms of set theory, group theory, and topology without introducing free parameters, ad-hoc axioms, or new invented entities; all claims rest on classical definitions of Alexandroff spaces, topological groups, and paratopological groups.

axioms (1)

standard math Standard axioms of group theory and point-set topology
Invoked throughout to define groups, continuity of operations, and the Alexandroff property.

pith-pipeline@v0.9.0 · 5449 in / 1308 out tokens · 56387 ms · 2026-05-07T11:48:19.316599+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 1 canonical work pages

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