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Level A Β· Machine-checkable Hard Geometry P-paper-weakly-first-countable

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.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) β€” original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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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 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

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 .

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). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem existsWeaklyFirstCountableCompactBig : answer(sorry) ↔
    βˆƒ (X : Type) (_ : TopologicalSpace X),
      WeaklyFirstCountableTopology X ∧ CompactSpace X ∧ T2Space X ∧
        𝔠 < #X
theorem existsWeaklyFirstCountableCompactNotFirstCountable :
    βˆƒ (X : Type) (_ : TopologicalSpace X),
      WeaklyFirstCountableTopology X ∧ CompactSpace X ∧ T2Space X ∧
        Β¬ FirstCountableTopology X
theorem cardinalMk_le_continuum_of_weaklyFirstCountable_of_countableSouslinNumber :
    answer(sorry) ↔ βˆ€ (X : Type) (_ : TopologicalSpace X), T35Space X β†’
      WeaklyFirstCountableTopology X β†’ HasCountableSouslinNumber X β†’ #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 β€” 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) β€” report it upstream too.

References

  • [Ar2013] Arhangeliski, Alexandr. "Selected old open problems in general topology." Buletinul Academiei de ŞtiinΕ£e 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 e6d1743831c2. Statements and descriptions Β© The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.