dense_sUnion_interior_of_closed — Mathlib

English

If countably many closed sets cover the whole space, then their interiors' union is dense.

Русский

Если счетное множество замкнутых множеств покрывает всю пространствo, то объединение их interiors плотное.

LaTeX

$$$IsGδ(\\\\text{univ}).dense_biUnion_interior_of_closed (dense_univ) \\text{ ... }$$$

Lean4

/-- Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with `⋃₀`. -/
theorem dense_sUnion_interior_of_closed {S : Set (Set X)} (hc : ∀ s ∈ S, IsClosed s) (hS : S.Countable)
    (hU : ⋃₀ S = univ) : Dense (⋃ s ∈ S, interior s) :=
  IsGδ.univ.dense_sUnion_interior_of_closed dense_univ hS hc hU.ge

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