Conjectures around homogeneous topological spaces

Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	5 open, 0 solved	Formal Conjectures

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Cite

@misc{cairn-paper-homogenous,
  title        = {Conjectures around homogeneous topological spaces},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-homogenous}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

homogeneousSpace_exists_inj_tendsto. Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?

homogeneousSpace_exists_surjective. Problem 14 in [Ar2013]: Is it possible to represent an arbitrary compact hausdorff space as an image of a homogeneous compact space under a continuous mapping?

firstCountableTopology_of_countablyMonolithicSpace. Problem 15 in [Ar2013]: Is every homogeneous ω-monolithic compact hausdorff space first countable?

countablyMonolithicSpace_card_lt. Problem 16 in [Ar2013]: Is the cardinality of every homogeneous ω-monolithic compact hausdorff space not greater than 𝔠?

monolithicSpace_exists_nhds_generated_countable. Problem 17 in [Ar2013]: Is it true that every nonempty monolithic compact hausdorff space contains a point with a first countable neighborhood basis?

Note: Nonempty X is required since the conclusion asserts the existence of a point.

This file formalizes the notions of a homogeneous topological space and of (ω-)monolithic topological spaces, and states some open problems about homogeneous and monolithic compact spaces.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.Homogenous (5 statements).

theorem homogeneousSpace_exists_inj_tendsto :
    ∀ (X : Type) (_ : TopologicalSpace X), ¬ Finite X → T2Space X → CompactSpace X →
      HomogeneousSpace X → ∃ s : ℕ → X, s.Injective ∧ ∃ a : X, Tendsto s atTop (nhds a)

theorem homogeneousSpace_exists_surjective :
    ∀ (X : Type) (_ : TopologicalSpace X), T2Space X → CompactSpace X →
      ∃ (Y : Type) (_ : TopologicalSpace Y), T2Space Y ∧ CompactSpace Y ∧ HomogeneousSpace Y ∧
        ∃ f : Y → X, Continuous f ∧ f.Surjective

theorem firstCountableTopology_of_countablyMonolithicSpace :
    ∀ (X : Type) (_ : TopologicalSpace X), T2Space X → CompactSpace X →
      HomogeneousSpace X → CountablyMonolithicSpace X → FirstCountableTopology X

theorem countablyMonolithicSpace_card_lt :
    ∀ (X : Type) (_ : TopologicalSpace X), T2Space X → CompactSpace X →
      HomogeneousSpace X → CountablyMonolithicSpace X → #X ≤ 𝔠

theorem monolithicSpace_exists_nhds_generated_countable :
    ∀ (X : Type) (_ : TopologicalSpace X), T2Space X → CompactSpace X →
      Nonempty X → MonolithicSpace X → ∃ x : X, (𝓝 x).IsCountablyGenerated

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[Ar2013] Arhangeliski, Alexandr. "Selected old open problems in general topology." Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica 73.2-3 (2013): 37-46. https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf
[Ar1987] Arhangel'skii, A. V. "Topological homogeneity. Topological groups and their continuous images." Russian Mathematical Surveys 42.2 (1987): 83-131. https://doi.org/10.1070/RM1987v042n02ABEH001333

Source and licence

Imported from Formal Conjectures (research papers), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.