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[Merged by Bors] - feat(Topology/UniformSpace): add UniformContinuous.iterate #14066

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[Merged by Bors] - feat(Topology/UniformSpace): add UniformContinuous.iterate #14066

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feat(Topology/UniformSpace): add `uniformContinuous_ite `
pitmonticone Jun 24, 2024
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Update Basic.lean
pitmonticone Jun 24, 2024
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Update Basic.lean
pitmonticone Jun 24, 2024
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pitmonticone Jun 24, 2024
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8 changes: 8 additions & 0 deletions Mathlib/Topology/UniformSpace/Basic.lean
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Expand Up @@ -1120,6 +1120,14 @@ nonrec theorem UniformContinuous.comp [UniformSpace β] [UniformSpace γ] {g :
hg.comp hf
#align uniform_continuous.comp UniformContinuous.comp

/--If a function `T` is uniformly continuous in a uniform space `β`,
then its `n`-th iterate `T^[n]` is also uniformly continuous.-/
theorem UniformContinuous.iterate [UniformSpace β] (T : β → β) (n : ℕ) (h : UniformContinuous T) :
UniformContinuous T^[n] := by
induction n with
| zero => exact uniformContinuous_id
| succ n hn => exact Function.iterate_succ _ _ ▸ UniformContinuous.comp hn h

theorem Filter.HasBasis.uniformContinuous_iff {ι'} [UniformSpace β] {p : ι → Prop}
{s : ι → Set (α × α)} (ha : (𝓤 α).HasBasis p s) {q : ι' → Prop} {t : ι' → Set (β × β)}
(hb : (𝓤 β).HasBasis q t) {f : α → β} :
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