Many coarse topologies on the real line

Gerald Kuba

arxiv: 2604.05949 · v1 · submitted 2026-04-07 · 🧮 math.GN

Many coarse topologies on the real line

Gerald Kuba This is my paper

Pith reviewed 2026-05-10 18:04 UTC · model grok-4.3

classification 🧮 math.GN

keywords Hausdorff topologiescoarser topologiesreal linecompletely normalBaire spacesfirst categorycompletely metrizabletopology lattices

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The pith

The real line admits 2^c pairwise non-homeomorphic completely normal topologies coarser than the usual one, split evenly into Baire and first-category examples.

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

The paper examines the family L of all Hausdorff topologies on the real numbers that are coarser than the standard Euclidean topology. It establishes that L contains 2^c many mutually non-homeomorphic completely normal topologies, with 2^c of them Baire spaces and 2^c of them spaces of first category. The work also produces c many pairwise non-homeomorphic completely metrizable topologies inside L. In addition, it studies the complete lattice structure on the topologies belonging to L and builds extremely long chains in which every topology is homeomorphic to the others.

Core claim

Let c denote the cardinality of the continuum. Let L denote the family of all Hausdorff topologies on the real line coarser than the natural topology. We construct 2^c pairwise non-homeomorphic completely normal topologies in L among which 2^c are Baire and 2^c are of first category. We also construct c pairwise non-homeomorphic completely metrizable topologies in L. Furthermore, we investigate complete lattices of topologies in L and construct extremely long chains of homeomorphic topologies in L.

What carries the argument

The family L of Hausdorff topologies on the reals coarser than the standard topology, together with explicit constructions that select different subsets of the reals to generate distinct neighborhood bases while preserving complete normality and the Baire or first-category property.

Load-bearing premise

There exist enough distinct subsets of the reals to define 2^c many different neighborhood systems on the line that stay Hausdorff, completely normal, and either Baire or first-category while remaining coarser than the usual topology.

What would settle it

A demonstration that every pair of completely normal Hausdorff topologies in L are homeomorphic, or that the total number of homeomorphism classes inside L is strictly less than 2^c, would refute the main construction.

read the original abstract

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

Kuba constructs 2^c non-homeomorphic completely normal coarse Hausdorff topologies on R in ZFC, split evenly into Baire and first-category families, with c metrizable examples and some lattice and chain results.

read the letter

The main takeaway is that this paper delivers ZFC constructions of 2^c pairwise non-homeomorphic completely normal topologies on the reals that are coarser than the usual topology. Of those, 2^c are Baire spaces and 2^c are first category, with a separate family of c non-homeomorphic completely metrizable ones. It also examines complete lattices of such topologies and produces very long chains of homeomorphic topologies within the family L. These cardinalities and the Baire/first-category split are the concrete new pieces. The work sits in the literature on coarse topologies and shows that the poset of Hausdorff topologies coarser than the standard one on R reaches the maximum possible size 2^c while keeping strong separation axioms. The lattice and chain results add some structural information about how these topologies sit together. The constructions appear to rely on selecting enough distinct subsets or functions on the reals to force non-homeomorphism while preserving the required properties like complete normality and the category distinctions. Staying inside ZFC without extra assumptions is a clear strength if the details check out. The soft spot is that the abstract states the existence and cardinalities but the full verification of property preservation in the constructions is not visible here. In set-theoretic topology those steps can hide small gaps around normality or category, so the claims rest on whether the proofs handle the distinctions cleanly. This paper is for people already working in general or set-theoretic topology who track the lattice of topologies on R or cardinal questions about non-homeomorphic examples. A reader in that niche would find the specific numbers and splits useful. It deserves a serious referee because the results are specific, new, and stated in ZFC, even though the proofs will need close checking.

Referee Report

0 major / 4 minor

Summary. The paper constructs, in ZFC, 2^c pairwise non-homeomorphic completely normal Hausdorff topologies on the real line that are coarser than the Euclidean topology, of which 2^c are Baire and 2^c are of first category. It further constructs c pairwise non-homeomorphic completely metrizable topologies in the same family L. The manuscript also studies complete lattices of topologies in L and produces extremely long chains of homeomorphic topologies within L.

Significance. If the constructions are correct, the results establish large cardinalities of distinct coarse topologies on ℝ with strong separation and category properties, together with structural information on the lattice and chains. The explicit ZFC constructions without additional axioms or fitted parameters constitute a clear strength, as does the focus on non-homeomorphism and preservation of complete normality, Baire/first-category, and metrizability.

minor comments (4)

[Introduction] The definition of the family L (Hausdorff topologies coarser than the natural topology) is stated in the abstract but should be repeated with full precision at the start of §1 or §2 to aid readability.
[Main construction section] In the construction of the 2^c completely normal topologies, the verification that the topologies remain coarser than the Euclidean topology and that non-homeomorphism is witnessed by specific invariants should be cross-referenced explicitly to the relevant lemmas.
[Lattice section] The lattice-theoretic results would benefit from a brief diagram or explicit description of the order relation used when discussing complete lattices of topologies in L.
[Throughout] A few instances of overloaded notation (e.g., reuse of symbols for different families of sets) appear; a short notation table or consistent subscripting would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contents, and the recommendation of minor revision. No specific major comments or requested changes were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is a pure existence paper in set-theoretic topology. It constructs (in ZFC) large families of distinct Hausdorff topologies on R coarser than the usual topology, with prescribed separation, category, and metrizability properties, plus lattice and chain results. The constructions rely on choosing sufficiently many distinct subsets or functions on the reals; they do not fit parameters to data, rename known empirical patterns, invoke self-citations as uniqueness theorems, or reduce any claimed result to an input by definition. The central claims are therefore independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies only on standard ZFC set theory and basic definitions from general topology; no free parameters, invented entities, or ad hoc axioms are introduced.

axioms (1)

standard math ZFC set theory
Cardinality calculations such as 2^c and existence of distinguishing families rely on the standard axioms of ZFC.

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discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Springer 1974

Comfort, W.W., and Negrepontis, S.: The Theory of Ultraﬁlters. Springer 1974

work page 1974
[2]

Heldermann 1989

Engelking, R.: General Topology , revised and completed edition. Heldermann 1989

work page 1989
[3]

Jech, Set Theory , 3rd ed

T. Jech, Set Theory , 3rd ed. Springer 2002

work page 2002
[4]

Kuba, G.: Counting metric spaces. Arch. Math. 97 , 569-578 (2011)

work page 2011
[5]

Matematiˇ cki Vesnik 64 (2), 125-137 (2012)

Kuba, G.: On certain separable and connected reﬁnements of the Euclidean topology. Matematiˇ cki Vesnik 64 (2), 125-137 (2012)

work page 2012
[6]

Internat

Kuba, G., On the variety of Euclidean point sets. Internat. Math. News 228 , 23-32 (2015) and arXiv:2004.11101v1

work page arXiv 2015
[7]

Kuba, G., Counting ultrametric spaces. Coll. Math. 152 , 217-234 (2018)

work page 2018
[8]

arXiv:2006.02880v1

Kuba, G.: Counting overweight spaces. arXiv:2006.02880v1

work page arXiv 2006
[9]

Lelek, A., and McAuley, L.F.: On hereditarily locally connected spaces and one-to-one continuous images of a line. Coll. Math. 17 , 319-324 (1967). Author’s address: Institute of Mathematics, BOKU University, Vienna. E-mail: gerald.kuba(at)boku.ac.at 19

work page 1967

[1] [1]

Springer 1974

Comfort, W.W., and Negrepontis, S.: The Theory of Ultraﬁlters. Springer 1974

work page 1974

[2] [2]

Heldermann 1989

Engelking, R.: General Topology , revised and completed edition. Heldermann 1989

work page 1989

[3] [3]

Jech, Set Theory , 3rd ed

T. Jech, Set Theory , 3rd ed. Springer 2002

work page 2002

[4] [4]

Kuba, G.: Counting metric spaces. Arch. Math. 97 , 569-578 (2011)

work page 2011

[5] [5]

Matematiˇ cki Vesnik 64 (2), 125-137 (2012)

Kuba, G.: On certain separable and connected reﬁnements of the Euclidean topology. Matematiˇ cki Vesnik 64 (2), 125-137 (2012)

work page 2012

[6] [6]

Internat

Kuba, G., On the variety of Euclidean point sets. Internat. Math. News 228 , 23-32 (2015) and arXiv:2004.11101v1

work page arXiv 2015

[7] [7]

Kuba, G., Counting ultrametric spaces. Coll. Math. 152 , 217-234 (2018)

work page 2018

[8] [8]

arXiv:2006.02880v1

Kuba, G.: Counting overweight spaces. arXiv:2006.02880v1

work page arXiv 2006

[9] [9]

Lelek, A., and McAuley, L.F.: On hereditarily locally connected spaces and one-to-one continuous images of a line. Coll. Math. 17 , 319-324 (1967). Author’s address: Institute of Mathematics, BOKU University, Vienna. E-mail: gerald.kuba(at)boku.ac.at 19

work page 1967