iUnion — Mathlib

English

If every s_i is separable, then the universal product over i is separable.

Русский

Если каждый s_i сепарабелен, то произведение по всем i сепарабельно.

LaTeX

$$$\forall ι\,[Countable\ ι],\forall X:ι\to Type*,\forall s: \forall i, Set(X_i),\ (\forall i, IsSeparable(s_i)) \Rightarrow IsSeparable(\text{univ.pi } s)$$$

Lean4

theorem iUnion {ι : Sort*} [Countable ι] {s : ι → Set α} (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) :=
  by
  choose c hc h'c using hs
  refine ⟨⋃ i, c i, countable_iUnion hc, iUnion_subset_iff.2 fun i => ?_⟩
  exact (h'c i).trans (closure_mono (subset_iUnion _ i))

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