isAtom_coe — Mathlib

English

In a T1 space, an atom of the lattice of closed sets is precisely a singleton closed set: IsAtom s iff there exists x with s = Closeds.singleton x.

Русский

В пространстве T1 атом в решётке замкнутых множеств равен precisely замкнутому множеству-одиночке: IsAtom s ⇔ ∃ x, s = Closeds.singleton x.

LaTeX

$$$ IsAtom(s) \\iff \\exists x, s = Closeds.singleton(x). $$$

Lean4

@[simp, norm_cast]
theorem isAtom_coe [T1Space α] {s : Closeds α} : IsAtom (s : Set α) ↔ IsAtom s :=
  Closeds.gi.isAtom_iff' rfl (fun t ht ↦ by obtain ⟨x, rfl⟩ := Set.isAtom_iff.1 ht; exact closure_singleton) s

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