Conjectures about Weakly First Countable spaces

Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space X such that 𝔠 < |X|. Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	3 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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Cite

@misc{cairn-paper-weakly-first-countable,
  title        = {Conjectures about Weakly First Countable spaces},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-weakly-first-countable}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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

existsWeaklyFirstCountableCompactBig. Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space X such that $c < ∣ X ∣$ .

Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.

existsWeaklyFirstCountableCompactNotFirstCountable. Problem 3 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space which is not first countable.

Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.

cardinalMk_le_continuum_of_weaklyFirstCountable_of_countableSouslinNumber. Problem 4 in [Ar2013]: If a Tychonoff weakly first-countable space has countable Souslin number, then does its cardinality not exceed the continuum?

This file formalizes the notion of a weakly first countable topological space and some conjectures around those.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.WeaklyFirstCountable (3 statements).

theorem existsWeaklyFirstCountableCompactBig :
    ∃ (X : Type) (_ : TopologicalSpace X),
      WeaklyFirstCountableTopology X ∧ CompactSpace X ∧ T2Space X ∧
        𝔠 < #X

theorem existsWeaklyFirstCountableCompactNotFirstCountable :
    ∃ (X : Type) (_ : TopologicalSpace X),
      WeaklyFirstCountableTopology X ∧ CompactSpace X ∧ T2Space X ∧
        ¬ FirstCountableTopology X

theorem cardinalMk_le_continuum_of_weaklyFirstCountable_of_countableSouslinNumber :
    ∀ (X : Type) (_ : TopologicalSpace X), T35Space X →
      WeaklyFirstCountableTopology X → HasCountableSouslinNumber X → #X ≤ 𝔠

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
[Ya1976] Yakovlev, N. N. "On the theory of o-metrizable spaces." Doklady Akademii Nauk. Vol. 229. No. 6. Russian Academy of Sciences, 1976. https://www.mathnet.ru/links/016f74007f9f96fa3aadae05cbd98457/dan40570.pdf (in Russian)

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.