feat(Paper): two conjectures about weakly first countable spaces #1860
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felixpernegger
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Jan 23, 2026
Paul-Lez
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Jan 23, 2026
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| /-- A topologocal space $X$ is called *weakly first countable* if there exists a function | ||
| $N : X → ℕ → Set X, such that: | ||
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| * For all $x : X, n : ℕ$ we have $x ∈ V x n$ | ||
| * For all $x : X, n : ℕ$: $V (n + 1) x ⊆ V n x$ | ||
| * $O ⊆ X$ is open iff $∀ x ∈ O, ∃ n : ℕ, V x n ⊆ O$ | ||
| -/ | ||
| class WeaklyFirstCountableTopology (X : Type*) [TopologicalSpace X] : Prop where | ||
| nhds_countable_weak_basis : ∃ V : X → ℕ → Set X, ∀ (x : X), Antitone (V x) ∧ ∀ (n : ℕ), x ∈ V x n ∧ | ||
| ∀ O : Set X, IsOpen O ↔ ∀ x ∈ O, ∃ k : ℕ, V x k ⊆ O | ||
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| /-- There are weakly first countable spaces which are not first countable, | ||
| for example the [Arens Space](https://topology.pi-base.org/spaces/S000156). -/ | ||
| @[category undergraduate, AMS 54] | ||
| theorem exists_weakly_first_countable_not_first_countable : ∃ (X : Type*) (_ : TopologicalSpace X), | ||
| WeaklyFirstCountableTopology X ∧ ¬ FirstCountableTopology X := by sorry | ||
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| /-- Every first countable space is weakly first countable, | ||
| simply take $N x$ as a countable neighborhood basis of $x$. -/ | ||
| @[category test, AMS 54] | ||
| instance FirstCountableTopology.weaklyFirstCountableTopology (X : Type*) [TopologicalSpace X] | ||
| [FirstCountableTopology X] : WeaklyFirstCountableTopology X where | ||
| nhds_countable_weak_basis := by sorry |
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I think these would be reasonable to add to FormalConjecturesForMathlib (with proof_wanted for statements that have a sorry - you can also keep those as problems in this file)
YaelDillies
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Jan 25, 2026
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Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>
Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>
YaelDillies
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Jan 25, 2026
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| * For all $x : X, n : ℕ$ we have $x ∈ V x n$ | ||
| * For all $x : X, n : ℕ$: $V (n + 1) x ⊆ V n x$ |
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Inconsistent order of arguments
| -/ | ||
| class WeaklyFirstCountableTopology (X : Type*) [TopologicalSpace X] : Prop where | ||
| nhds_countable_weak_basis : ∃ V : X → ℕ → Set X, ∀ (x : X), Antitone (V x) ∧ ∀ (n : ℕ), x ∈ V x n ∧ | ||
| ∀ O : Set X, IsOpen O ↔ ∀ x ∈ O, ∃ k : ℕ, V x k ⊆ O |
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Suggested change
| ∀ O : Set X, IsOpen O ↔ ∀ x ∈ O, ∃ k : ℕ, V x k ⊆ O | |
| (\nhds x).HasBasis (fun _ \mapsto True) (V x) |
| @[category test, AMS 54] | ||
| instance FirstCountableTopology.weaklyFirstCountableTopology (X : Type*) [TopologicalSpace X] | ||
| [FirstCountableTopology X] : WeaklyFirstCountableTopology X where | ||
| nhds_countable_weak_basis := by sorry |
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Problem 3 and 4 of https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf
See #1849.
I guess weakly first countable could be defined better, but maybe this is enough for this repo.