hull_kernel_of_isClosed — Mathlib

English

If T order-generates and every p ∈ T is InfPrime, then for any closed C ⊆ T we have hull(T, kernel(C)) = C.

Русский

Если T порождает порядок и все p ∈ T удовлетворяют InfPrime, то для любого замкнутого C ⊆ T верно hull(T, kernel(C)) = C.

LaTeX

$$$\\text{If } hG, hT \\text{ hold and } C \\subseteq T \\text{ is closed, then } \\mathrm{hull}(T, \\ker(C)) = C.$$$

Lean4

theorem hull_kernel_of_isClosed [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T)
    {C : Set T} (h : IsClosed C) : hull T (kernel C) = C :=
  by
  obtain ⟨a, ha⟩ := (isClosed_iff hT).mp h
  rw [ha, kernel_hull hG]

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