secondCountableTopology_of_countable

English

If α has discrete topology and is countable, then α is second-countable.

Русский

Если у α дискретная топология и множество счётно, то она الثانية по счётности (second-countable).

LaTeX

$$∀ α, [DiscreteTopology α] [Countable α] → SecondCountableTopology α$$

Lean4

instance (priority := 100) secondCountableTopology_of_countable [hd : DiscreteTopology α] [Countable α] :
    SecondCountableTopology α :=
  haveI : ∀ i : α, SecondCountableTopology (↥({ i } : Set α)) := fun i =>
    { is_open_generated_countable := ⟨{ univ }, countable_singleton _, by simp only [eq_iff_true_of_subsingleton]⟩ }
  secondCountableTopology_of_countable_cover (fun _ ↦ isOpen_discrete _) (iUnion_of_singleton α)

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