dense — Mathlib

Lean4
/-- The dense Grothendieck topology.

See https://ncatlab.org/nlab/show/dense+topology, or [MM92] Chapter III, Section 2, example (e).
-/
def dense : GrothendieckTopology C
    where
  sieves X S := ∀ {Y : C} (f : Y ⟶ X), ∃ (Z : _) (g : Z ⟶ Y), S (g ≫ f)
  top_mem' _ Y _ := ⟨Y, 𝟙 Y, ⟨⟩⟩
  pullback_stable' := by
    intro X Y S h H Z f
    rcases H (f ≫ h) with ⟨W, g, H'⟩
    exact ⟨W, g, by simpa⟩
  transitive' := by
    intro X S H₁ R H₂ Y f
    rcases H₁ f with ⟨Z, g, H₃⟩
    rcases H₂ H₃ (𝟙 Z) with ⟨W, h, H₄⟩
    exact ⟨W, h ≫ g, by simpa using H₄⟩
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