English
If δ1, δ2 and φ1, φ2 are proper maps whose ranges cover α, then the uniform structure on C(α,β) is the infimum of the uniform structures induced by precomposition with φ1 and φ2.
Русский
Если φ1, φ2 — правильные отображения с покрытиями, то униформность на C(α,β) равна infimum униформностей, полученных пред-композицией с φ1 и φ2.
LaTeX
$$$\\text{UniformSpace } C(α, β) = \\text{inf}_{\} \\big( \\text{comap}(\\,\\cdot, φ1), \\text{comap}(\\cdot, φ2) \\big)$ under appropriate hypotheses$$
Lean4
/-- Given functions `F i, f` which are continuous on a compact set `s`, `F` tends to `f`
uniformly on `s` if and only if the restrictions (as elements of `C(s, β)`) converge. -/
theorem _root_.ContinuousOn.tendsto_restrict_iff_tendstoUniformlyOn {s : Set α} [CompactSpace s] {f : α → β}
(hf : ContinuousOn f s) {ι : Type*} {p : Filter ι} {F : ι → α → β} (hF : ∀ i, ContinuousOn (F i) s) :
Tendsto (fun i ↦ ⟨_, (hF i).restrict⟩ : ι → C(s, β)) p (𝓝 ⟨_, hf.restrict⟩) ↔ TendstoUniformlyOn F f p s :=
by
rw [ContinuousMap.tendsto_iff_tendstoUniformly, tendstoUniformlyOn_iff_tendstoUniformly_comp_coe]
congr!
