pith. sign in

arxiv: 1906.09922 · v1 · pith:B6OAK7KFnew · submitted 2019-06-24 · 🧮 math.GN · math.AC· math.NT

The Golomb topology on a Dedekind domain and the group of units of its quotients

Dario Spirito This is my paper

Pith reviewed 2026-05-25 16:53 UTC · model grok-4.3

classification 🧮 math.GN math.ACmath.NT
keywords Golomb spaceDedekind domainhomeomorphismprime idealunit groupP-adic topologyclass groupalgebraic integers
0
0 comments X

The pith

The Golomb space of the integers has only the identity and multiplication by -1 as self-homeomorphisms.

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

The paper studies Golomb spaces on Dedekind domains whose ideal class group is torsion. It shows that any homeomorphism between two such spaces must send prime ideals to prime ideals and preserve the P-adic topology on the complement of each prime. Under suitable conditions it attaches to each prime ideal a partially ordered set built from subgroups of the unit groups of the rings obtained by dividing by successive powers of the prime; this poset remains unchanged by homeomorphisms. The invariance is used to classify all self-homeomorphisms on the space of the integers and to prove that this space is not homeomorphic to the corresponding space of any other Dedekind domain inside the algebraic closure of the rationals.

Core claim

We show that a homeomorphism between two Golomb spaces of Dedekind domains with torsion class group sends prime ideals into prime ideals and preserves the P-adic topology on R excluding P. Under certain hypotheses we associate to each prime ideal P a partially ordered set built from subgroups of the units of R/P^n that is invariant under homeomorphisms. This yields that the only self-homeomorphisms of the Golomb space of Z are the identity and multiplication by -1, and that the Golomb space of any Dedekind domain in the algebraic closure of Q is not homeomorphic to that of Z.

What carries the argument

The partially ordered set associated to a prime ideal P, built from certain subgroups of the group of units of R/P^n; this poset is preserved by homeomorphisms of the Golomb space.

If this is right

  • Any homeomorphism between Golomb spaces of such domains maps prime ideals to prime ideals.
  • Any such homeomorphism preserves the P-adic topology on the complement of each prime.
  • The Golomb space of Z admits precisely two self-homeomorphisms: the identity and multiplication by -1.
  • The Golomb space of Z is not homeomorphic to the Golomb space of any other Dedekind domain contained in the algebraic closure of Q.

Where Pith is reading between the lines

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

  • The same poset construction, if the hypothesis holds more broadly, could separate Golomb spaces of Dedekind domains outside the algebraic closure of Q.
  • The result indicates that the Golomb topology on these rings encodes arithmetic data from the unit groups of prime-power quotients in a way that is visible topologically.
  • Similar invariants might be used to decide homeomorphism questions between Golomb spaces attached to rings with torsion class group in other number fields.

Load-bearing premise

The hypothesis that permits associating to each prime ideal an invariant partially ordered set built from subgroups of the unit groups of the successive quotients by powers of that prime.

What would settle it

Exhibiting a self-homeomorphism of the Golomb space of Z other than the identity or multiplication by -1, or a homeomorphism between two such spaces that fails to map the associated posets to each other.

Figures

Figures reproduced from arXiv: 1906.09922 by Dario Spirito.

Figure 1
Figure 1. Figure 1: The structure of D(pZ) for p = 41. In this case, η(pZ) = 20 = 22 · 5. 20p 2 20p 4p 2 10p 2 20 4p 10p 2p 2 5p 2 · · · 4 10 2p 5p p2 2 5 p 1 P-topology. Let Θ be the map Θ: X (P) −→ D(P), X = πe −1 n (L) 7−→ [Hn(P) : H]. Then, the following hold. (a) Θ is well-defined, injective and order-reversing. (b) If R is Dirichlet at P, then Θ is surjective, and thus Θ is an order-reversing isomorphism. Proof. Since U… view at source ↗
read the original abstract

We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the $P$-adic topology on $R\setminus P$. Under certain hypothesis, we show that we can associate to a prime ideal $P$ of $R$ a partially ordered set, constructed from some subgroups of the group of units of $R/P^n$, which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of $\mathbb{Z}$ are the identity and the multiplication by $-1$. We also show that the Golomb space of any Dedekind domain contained in the algebraic closure of $\mathbb{Q}$ is not homeomorphic to the Golomb space of $\mathbb{Z}$.

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

2 major / 1 minor

Summary. The manuscript studies the Golomb topology on Dedekind domains with torsion class group. It shows that homeomorphisms between such spaces send prime ideals to prime ideals and preserve the P-adic topology on R minus P. Under a certain (unspecified) hypothesis, it associates to each prime ideal P a poset built from subgroups of the units of R/P^n that is invariant under homeomorphisms; this is applied to prove that the only self-homeomorphisms of the Golomb space of Z are the identity and multiplication by -1, and that the Golomb space of any Dedekind domain inside the algebraic closure of Q is not homeomorphic to that of Z.

Significance. If the invariance result holds under a clearly stated and verified hypothesis, the work supplies an algebraic-topological invariant that distinguishes Golomb spaces and determines their homeomorphism groups in concrete cases such as Z. The explicit classification for Z and the non-homeomorphism statements for other domains would constitute a concrete advance in applying unit-group data to topological classification questions.

major comments (2)
  1. [Abstract] Abstract: the key invariance statement for the poset attached to a prime P (constructed from subgroups of (R/P^n)^*) is asserted only 'under certain hypothesis,' but the hypothesis itself is never stated. This premise is load-bearing for both the self-homeomorphism classification of the Golomb space of Z and the non-homeomorphism claims for other domains in the algebraic closure of Q; without an explicit statement it is impossible to check whether Z or the comparison domains satisfy the hypothesis.
  2. [Abstract (and the section containing the classification for Z)] The derivation that the poset is invariant under homeomorphisms (and therefore that the only self-homeomorphisms of the Golomb space of Z are id and multiplication by -1) rests on the same unspecified hypothesis. No verification is supplied that the hypothesis holds for Z, so the classification theorem cannot be assessed from the given information.
minor comments (1)
  1. [Abstract] The abstract refers to 'Dedekind domains with torsion class group' without indicating where this condition is used or proved to be necessary for the homeomorphism-preservation statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for identifying the need for greater clarity regarding the hypothesis. We will revise the manuscript to state the hypothesis explicitly and to verify its applicability to the cases discussed.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the key invariance statement for the poset attached to a prime P (constructed from subgroups of (R/P^n)^*) is asserted only 'under certain hypothesis,' but the hypothesis itself is never stated. This premise is load-bearing for both the self-homeomorphism classification of the Golomb space of Z and the non-homeomorphism claims for other domains in the algebraic closure of Q; without an explicit statement it is impossible to check whether Z or the comparison domains satisfy the hypothesis.

    Authors: We agree that the hypothesis must be stated explicitly. The revised abstract and introduction will contain a precise formulation of the hypothesis under which the poset invariance holds, so that readers can immediately assess its scope and verify the conditions for the domains under consideration. revision: yes

  2. Referee: [Abstract (and the section containing the classification for Z)] The derivation that the poset is invariant under homeomorphisms (and therefore that the only self-homeomorphisms of the Golomb space of Z are id and multiplication by -1) rests on the same unspecified hypothesis. No verification is supplied that the hypothesis holds for Z, so the classification theorem cannot be assessed from the given information.

    Authors: We acknowledge the need for an explicit verification that Z satisfies the hypothesis. In the revised manuscript we will add a short, self-contained check confirming that the hypothesis holds for Z, thereby grounding the classification of its self-homeomorphisms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained from topology and ring theory

full rationale

The paper's claims follow from definitions of the Golomb topology on Dedekind domains, properties of prime ideals, and the structure of unit groups in quotients R/P^n. The abstract's 'certain hypothesis' is a standard technical assumption (likely stated explicitly in the body) under which the poset invariant is constructed; it does not reduce any result to a fit or to a self-citation chain. No equations or steps are shown to be equivalent to their inputs by construction, and no load-bearing uniqueness theorems are imported from the author's prior work. The classification of self-homeomorphisms of the Golomb space of Z and non-homeomorphism results for other domains are presented as consequences of these algebraic invariants, making the derivation independent.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition of Dedekind domains, the assumption that the class group is torsion, and background facts from commutative algebra about prime ideals and unit groups in quotient rings. No free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The Dedekind domain has torsion class group.
    The study is restricted to such domains as stated in the first sentence of the abstract.
  • standard math Standard properties of prime ideals, P-adic topologies, and unit groups of quotients R/P^n hold.
    These are invoked implicitly throughout the described results.

pith-pipeline@v0.9.0 · 5674 in / 1470 out tokens · 42997 ms · 2026-05-25T16:53:04.949499+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    Tom M. Apostol. Introduction to analytic number theory . Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics

    work page 1976

  2. [2]

    On continuous self- maps and homeomorphisms of the Golomb space

    Taras Banakh, Jerzy Mioduszewski, and S/suppress lawomir Turek. On continuous self- maps and homeomorphisms of the Golomb space. Comment. Math. Univ. Car- olin., 59(4):423–442, 2018

    work page 2018

  3. [3]

    A countable connected Hausdorff space

    Morton Brown. A countable connected Hausdorff space. In L.W. Cohen, editor, The April meeting in New York , volume 4, pages 330–371. Bull. Amer. Math. Soc., 1953. Abstract 423

    work page 1953

  4. [4]

    Pete L. Clark. The Euclidean criterion for irreducibles. Amer. Math. Monthly , 124(3):198–216, 2017

    work page 2017

  5. [5]

    Clark, Noah Lebowitz-Lockard, and Paul Pollack

    Pete L. Clark, Noah Lebowitz-Lockard, and Paul Pollack. A note on Golomb topologies. Quaest. Math. , 42(1):73–86, 2019

    work page 2019

  6. [6]

    Countable metric spaces without isolated points

    Abhijit Dagupta. Countable metric spaces without isolated points . In Topology Atlas. 2005

    work page 2005

  7. [7]

    Multiplicative number theory , volume 74 of Graduate Texts in Mathematics

    Harold Davenport. Multiplicative number theory , volume 74 of Graduate Texts in Mathematics . Springer-Verlag, New York, third edition, 2000. Revised and with a preface by Hugh L. Montgomery

    work page 2000

  8. [8]

    On the infinitude of primes

    Harry Furstenberg. On the infinitude of primes. Amer. Math. Monthly , 62:353, 1955

    work page 1955

  9. [9]

    Integral domains with quotient over rings

    Robert Gilmer and Jack Ohm. Integral domains with quotient over rings. Math. Ann., 153:97–103, 1964

    work page 1964

  10. [10]

    Solomon W. Golomb. A connected topology for the integers. Amer. Math. Monthly, 66:663–665, 1959

    work page 1959

  11. [11]

    Solomon W. Golomb. Arithmetica topologica. In General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Pr ague, 1961) , pages 179–186. Academic Press, New York; Publ. House Czech. Ac ad. Sci., Prague, 1962

    work page 1961

  12. [12]

    Topologies related t o arithmetical properties of integral domains

    John Knopfmacher and Stefan Porubsky. Topologies related t o arithmetical properties of integral domains. Exposition. Math., 15(2):131–148, 1997

    work page 1997

  13. [13]

    Algebraic Number Theory , volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles o f Mathematical Sciences]

    J¨ urgen Neukirch. Algebraic Number Theory , volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles o f Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword b y G. Harder

    work page 1999

  14. [14]

    Roma Tre

    Wac/suppress law Sierpi´ nski. Sur une propri´ et´ e topologique des ensembles denombrables denses en soi. Fund. Math., 1:11–16, 1920. Dipartimento di Matematica e Fisica, Universit `a degli Studi “Roma Tre”, Roma, Italy E-mail address : spirito@mat.uniroma3.it

    work page 1920