isLowerSet_iff_nhds — Mathlib

English

Topology.IsLowerSet α is equivalent to the property that for every a, nhds a equals the principal of Iic a.

Русский

Topology.IsLowerSet α эквивалентно свойству, что для каждого a окрестности a равныPrincipal(Iic a).

LaTeX

$$$$ \\text{Topology.IsLowerSet }(\\alpha) \\quad\\Longleftrightarrow\\quad \\forall a,\\; \\mathcal{N}(a) = \\mathcal{P}(\\mathrm{Iic}(a)). $$$$

Lean4

protected theorem _root_.Topology.isLowerSet_iff_nhds {α : Type*} [TopologicalSpace α] [Preorder α] :
    Topology.IsLowerSet α ↔ (∀ a : α, 𝓝 a = 𝓟 (Iic a))
    where
  mp _ a := nhds_eq_principal_Iic a
  mpr hα := ⟨by simp [TopologicalSpace.ext_iff_nhds, hα, nhds_eq_principal_Iic]⟩

Похожие утверждения