intermediate_value₂_eventually₁

English

The closure operator from the closed-sets Galois connection coincides with hull(kernel(S)) when viewed via the duality.

Русский

Замыкание из той же Галуа связи совпадает с hull(kernel(S)) при учёте двойственности.

LaTeX

$$$gc\\_closureOperator(S) = hull(T, kernel(S)).$$$

Lean4

theorem intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s)
    [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (ha' : f a ≤ g a)
    (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x :=
  by
  rw [continuousOn_iff_continuous_restrict] at hf hg
  obtain ⟨b, h⟩ :=
    @intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ _
      (comap_coe_neBot_of_le_principal hl) _ _ hf hg ha' (he.comap _)
  exact ⟨b, b.prop, h⟩

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