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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | import OAI.Probability.InvariantIsing.Spectral.ResolventMeasureUniqueness
import Mathlib.MeasureTheory.Measure.Prokhorov
namespace OAI
/-! Finitely many separating tests control a continuous image of a compact space. -/
noncomputable section
open Set Filter MeasureTheory
open scoped Topology
namespace InvariantIsing
theorem compact_finite_test_control {X ι Y : Type*} [TopologicalSpace X] [CompactSpace X]
[PseudoMetricSpace Y] (F : ι → X → ℝ) (hF : ∀ i, Continuous (F i))
(hsep : ∀ x y, (∀ i, F i x = F i y) → x = y)
(G : X → Y) (hG : Continuous G) (x₀ : X) {ε : ℝ} (hε : 0 < ε) :
∃ s : Finset ι, ∃ δ : ι → ℝ, (∀ i, 0 < δ i) ∧
∀ x, (∀ i ∈ s, dist (F i x) (F i x₀) < δ i) → dist (G x) (G x₀) < ε := by
classical
let E : X → (ι → ℝ) := fun x i => F i x
have hE : Continuous E := continuous_pi hF
have he : Function.Injective E := fun x y h => hsep x y (congrFun h)
have hi := (hE.isClosedEmbedding he).isEmbedding.isInducing
have hu : G ⁻¹' Metric.ball (G x₀) ε ∈ 𝓝 x₀ :=
hG.continuousAt.preimage_mem_nhds (Metric.ball_mem_nhds _ hε)
rw [hi.nhds_eq_comap] at hu
obtain ⟨V,hV,hVU⟩ := Filter.mem_comap.mp hu
rw [nhds_pi] at hV
obtain ⟨s,W,hW,hWV⟩ := Filter.mem_pi'.mp hV
choose δ hδ hd using fun i => Metric.mem_nhds_iff.mp (hW i)
refine ⟨s,δ,hδ,fun x hx => ?_⟩
apply hVU
apply hWV
intro i hi'
exact hd i (hx i hi')
end InvariantIsing
end
end OAI
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