English
If a countable family of closed sets covers the whole space, then the union of their interiors is dense.
Русский
Если счетное семейство замкнутых множеств покрывает всю площадь, то объединение их interiors плотное.
LaTeX
$$$$IsGδ(\\\\text{univ}) \\\\Rightarrow Dense(\\\\bigcup t \\\\operatorname{interior}(t))$$$$
Lean4
/-- Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with an index set which is a countable set in any type. -/
theorem dense_biUnion_interior_of_closed {S : Set α} {f : α → Set X} (hc : ∀ s ∈ S, IsClosed (f s)) (hS : S.Countable)
(hU : ⋃ s ∈ S, f s = univ) : Dense (⋃ s ∈ S, interior (f s)) :=
IsGδ.univ.dense_biUnion_interior_of_closed dense_univ hS hc hU.ge
